Computer simulations of glasses: the potential

energy landscape

Zamaan Raza, Björn Alling and Igor Abrikosov

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

Zamaan Raza, Björn Alling and Igor Abrikosov, Computer simulations of glasses: the potential

energy landscape, 2015, Journal of Physics: Condensed Matter, (27), 29, 293201.

http://dx.doi.org/10.1088/0953-8984/27/29/293201

Copyright: IOP Publishing: Hybrid Open Access

http://www.iop.org/

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-120738

Topical Review

Computer simulations of glasses: the potentialenergy landscape

Zamaan Raza1, Bjorn Alling1 and Igor A. Abrikosov1,2,3

1 Department of Physics, Biology and Chemistry (IFM), Linkoping University,SE-581 83, Linkoping, Sweden2 Materials Modeling and Development Laboratory, NUST “MISIS”, 119049Moscow, Russia3 LOCOMAS Laboratory, Tomsk State University, 634050 Tomsk, Russia

E-mail: zamaan.raza@gmail.com

Abstract. We review the current state of research on glasses, discussing thetheoretical background and computational models employed to describe them.This article focuses on the use of the potential energy landscape (PEL) paradigmto account for the phenomenology of glassy systems, and the way in which it can beapplied in simulations and the interpretation of their results. This article providesa broad overview of the rich phenomenology of glasses, followed by a summaryof the theoretical frameworks developed to describe this phenomonology. Wediscuss the background of the PEL in detail, the onerous task of how to generatecomputer models of glasses, various methods of analysing numerical simulations,and the literature on the most commonly used model systems. Finally, we tacklethe problem of how to distinguish a good glass former from a good crystal formerfrom an analysis of the PEL. In summarising the state of the potential energylandscape picture, we develop the foundations for new theoretical methods thatallow the ab initio prediction of the glass-forming ability of new materials byanalysis of the PEL.Keywords: Glasses, Potential Energy Landscape, Computer Simulation

PACS numbers: 61.43.-j,61.43.Fs,64.70.kj,64.70.Q-

Submitted to: J. Phys.: Condens. Matter

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1 Introduction 21.1 Motivation . . . . . . . . . . . . . . . . 21.2 Definition of a glass . . . . . . . . . . . 31.3 Strong and fragile liquids . . . . . . . . 41.4 The glass transition . . . . . . . . . . . 51.5 The Kauzmann paradox . . . . . . . . . 6

2 Models of glasses 72.1 Mode-coupling theory (MCT) . . . . . . 72.2 Random first-order transition theory

(RFOT) . . . . . . . . . . . . . . . . . . 82.3 The potential energy landscape (PEL) . 92.4 The free energy landscape . . . . . . . . 122.5 Discussion . . . . . . . . . . . . . . . . . 12

3 Theoretical background 133.1 Thermodynamic formalism . . . . . . . 133.2 Density of configurational states and

entropy . . . . . . . . . . . . . . . . . . 143.3 Basin free energy . . . . . . . . . . . . . 15

4 Sampling schemes 154.1 Reverse Monte Carlo (RMC) . . . . . . 154.2 Molecular dynamics: aging versus equi-

librium . . . . . . . . . . . . . . . . . . . 154.3 Density of states Monte Carlo . . . . . . 184.4 Continuous Random Networks (CRN) . 184.5 Stochastic Quenching (SQ) . . . . . . . 184.6 Activation-Relaxation Technique (ART) 194.7 Discussion . . . . . . . . . . . . . . . . . 19

5 Theoeretical characterisation 195.1 Structure and packing . . . . . . . . . . 205.2 Density of states . . . . . . . . . . . . . 215.3 Normal mode analysis . . . . . . . . . . 215.4 Activated dynamics . . . . . . . . . . . 225.5 Network topology . . . . . . . . . . . . . 235.6 Discussion . . . . . . . . . . . . . . . . . 24

6 Model Systems 256.1 The binary Lennard-Jones (BLJ) glass . 256.2 Amorphous silica . . . . . . . . . . . . . 276.3 Hard sphere model . . . . . . . . . . . . 286.4 Discussion . . . . . . . . . . . . . . . . . 28

7 Ab initio models 29

8 Glass-forming ability 29

9 Conclusions 30

1. Introduction

1.1. Motivation

One of the biggest challenges in condensed matterphysics is the modelling of amorphous solids, whichlack the long range order characteristic of crystallinesolids. When modelling bulk crystals, it is possibleto exploit the translational symmetry of the system,allowing the construction of a unit cell subject toperiodic boundary conditions, i.e. the translation ofa basis along lattice points. This method does nottrivially map to amorphous phases which contain nosuch symmetries, although a large enough unit cellwill capture to some extent the short and mediumrange disorder in spite of the artificial periodicboundary conditions. This, however, comes at greatcomputational expense. For example, first principlescalculations in the framework of density functionaltheory scale as N3 for a unit cell containing N atoms,while the scaling of higher level quantum mechanicalmethods is even worse. For reasons elucidated below,interest in glasses is increasing, and with it the needfor a complete theoretical model, which is not currentlyavailable.

One such reason is the new class of materialscalled bulk metallic glasses. Metal atoms haveweak orientational constraints in the solid state, andtherefore tend to crystallise easily. Certain alloycompositions, such as Cu64Zr36 have a much lowertendency to crystallise, i.e. a good glass-forming ability[1], while others, such as Ni64Zr36 tend to form orderedstructures [2]. Bulk metallic glasses are currentlyof particular interest due to their unique physicalproperties, such as high elastic moduli or plasticity [3]and the potential for surface smoothness not limitedto crystal facets. They lack an intrinsic length scale(grain size in the case of crystalline structures, or chainlength in the case of polymer glasses), and can inprinciple be shaped at the atomic level [4]. Findingcompositions with good glass forming ability is a time-intensive task, with one composition taking around aday to fabricate; this trial-and-error process could begreatly streamlined using modelling methods [4].

Another reason is the development of physicalvapour deposition as a technique for synthesisingglasses with extraordinary thermodynamic and kineticstability [5–8]. Until recently, such ultrastable glasseswere limited to complex molecular and polymer glasses;more recently however, vapour deposition has allowedthe preparation of ultrastable metallic glasses [9].When formed from the liquid phase, the propertiesof a glass depend strongly on its thermal history,i.e. its cooling rate, and it cannot be assumed tobe at equilibrium, since glasses are known to “age”,or find more thermodynamically favourable states as

3

time passes. Recent developments in vapour depositiontechniques have allowed the synthesis of ultrastableglasses occupying equilibrium states equivalent to asupercooled liquid aged for thousands of years or cooledat a rate 19 orders of magnitude slower than currentmethods allow [10]. Although aging is believed toallow a glass to find more stable states, experimentaldeterminations of the enthalpies of ultrastable metallicglasses formed by vapour deposition suggests thattheir stability is not necessarily a result of beingin the lowest thermodynamic state [9]. There aremany empirical measures of glass forming ability (seesection 8), but there is not coherent microscopicstructural explanation that accounts for the differencesbetween compositions. Furthermore, it is possible tosynthesise stable glasses from compositions with lowphenomenological glass forming ability such as Fe1-xCx

using thin film vapour deposition techniques[11]. Thissuggests that there is something missing in the variousempirical measures of glass forming ability, and a morefundamental description is required.

An example of a good glass former is B2O3. AB2O3 melt does not crystallise even when seeded witha crystallite of the stable phase, as a result of thelow driving force and slow kinetics; its crystallisationcan only be induced at high pressure, when enthalpicconsiderations force the structure to adopt an optimalpacking [12]. This leads to the question of whichparameters define a good glass former. In general,we describe a material that has a high tendency toform a glass instead of a crystal when quenched.There are many empirical measures of glass-formingability [13–15], but none are derived from an atomisticdescription of a glass or offer a physical insight intothe structure of a glass. What is the driving force inthe glass transition? Is it possible to characterise agood glass former by the potential energy landscapealone, or do free energy considerations need to betaken into account? There is strong evidence thatcomposition and stoichiometry control glass formingability rather than simple physical and topologicalconsiderations [16]. Is there a parameter that can beused to theoretically distinguish a good glass formingcomposition from a crystal forming composition?Which variable, if any, describes glass forming ability?In this review, we take a step towards answeringthese questions, and in doing so, demonstrate thatthe potential energy landscape is a useful paradigm forpredicting the characteristics of amorphous systems.

The potential energy landscape (PEL) is a simpleconcept: for a system of N , atoms, the potentialenergy is defined for each atomic configuration, i.e.at each point of the 3N -dimensional configurationspace. The resulting hypersurface contains minima andsaddle points, at which the gradient of the potential

energy (the forces) are zero. The minima correspondto stable and metastable atomic configurations, andthe saddle points to transition states between minima.Common atomistic calculations examine local featuresof the landscape, such as a minimum (zero temperaturestructure optimisation) or a limited region determinedby the temperature (molecular dynamics), but our aimis to link collective and statistical features of the PELwith the properties of amorphous materials and glasses.

This review is organised as follows. In section 1we introduce the topic of computational modelling ofglasses and the problems therein, setting some basicdefintions. In section 2, we introduce the theoreticalframeworks most commonly used to model glasses,with an emphasis on the potential energy landscape.In section 3 we discuss the construction of the potentialenergy landscape paradigm in a more rigorous way,focusing on the statistical mechanics required tolink microscopic computer models to thermodynamicquantities. In section 4, we summarise the methodsfor sampling the potential energy landscape, i.e. howto generate atomistic models of glasses. In section 5we describe the methods by which such structuralmodels can be analysed. Simple empirical modelsof glasses are discussed in section 6, followed byan extension to the use of first principles electronicstructure calculations for higher quality models insection 7. Finally, we discuss possible methodsand parameters for distinguishing good crystal-formersfrom good glass-formers in section 8 and finish withsome conclusions in section 9.

1.2. Definition of a glass

A glass is an amorphous material for which, below acrossover temperature, the timescale of its dynamicalproperties suddenly and dramatically increase beyondthe experimental timescale. Below the transitiontemperature, all molecular motion ceases, except forthermal vibrations [17] (this is not strictly true, acounterexample being long timescale Johari-Goldsteinβ-processes [18] — see section 3.1; however, this is notenough to induce a viscous flow). There are numerouspossible definitions for the crossover temperature,discussed in section 1.4.

Generally, one can distinguish between twotypes of structurally (as opposed to magnetically)disordered system [19]. The first type is systemswith configurational but not structural disorder, suchas random alloys, in which atomic positions aretopologically fixed by the underlying Bravais lattice,but the distribution of atoms on the lattice israndom. Crystalline systems containing defects suchas vacancies, interstitial atoms and dislocations alsobelong to this category, since removal of these defectswill restore a disordered material to a crystalline

4

state. Atomic vibrations and local atomic relaxationsresult in displacements from the ideal lattice positions,but they preserve the translational symmetry of theunderlying Bravais lattice “on average,” and the theoryof such systems can be built using the usual formalismof reciprocal space [20]. The second type is comprisedof structurally or topologically disordered systems,where atomic positions cannot be associated with anyunderlying Bravais lattice. A straight line through aperfectly amorphous material in an arbitrary directionwill be divided by the atoms into “irregular intervalsof random sequence” [21].

A snapshot of the instantaneous atomic configura-tion of any amorphous solid is indistinguishable fromthat of a liquid. The glass transition is often attributedto a dynamic heterogeneity [22], since there has beenlittle evidence of any change in structure as indicatedby the static structure factor S(q) across the glass tran-sition temperature (although this is not true for somesystems [23]). However, a recent study suggests thatthe dynamic heterogeneity may have an origin in Ising-like criticality in static structural order [24]. A glassdoes not flow like a liquid, but its diffusive process arevery different to those that occur in crystals [25]. In-stead of single-atom hops, atoms in glasses undergocomplicated collective motions of ten or more atoms[26]. The diffusive processes can be divided into threetypes of local excitations: tunneling systems, excess vi-brations and local relaxations [25]. Within the soft po-tential model whichis parameterised using experimen-tal data, these effects are included in collective motionsof 20 or more atoms [27].

The difference between an amorphous solid anda glass is debatable since the terms are often usedinterchangeably in the literature. When a liquidis cooled below its experimental glass transitiontemperature, Tg, the timescale of its dynamicalproperties of a liquid suddenly and drastically increase.Its transport properties diverge from the Arrheniuslaw, for which the rate of a process is proportional toe−Ea/kBT where Ea is some activation energy. BelowTg, the viscosity of the system is above approximately1013P, and the system can no longer reach thermalequilibrium since the relaxation timescale exceeds theexperimental timescale (of the order 100 s) [28]. Aglass can therefore be considered to be a special caseof amorphous solid that undergoes the glass transitionwhen the liquid phase is cooled.

According to Ediger and Harrowell [29], there arethree characteristics of glasses formed from supercooledliquids (liquids cooled below their melting temperaturewithout undergoing crystallisation): firstly a hugeincrease in shear viscosity, secondly a temperaturedependence of the entropy manifested in the so-called Kauzmann paradox [30], and thirdly dynamic

heterogeneity, suggesting some kind of solid-liquidcoexistence, a more recent addition to menagerie ofglass theories [31].

Another definition is the transition from anergodic state to a non-ergodic state, i.e. from aliquid state which will in principle explore the entireconfigurational space given enough time, to a glassystate which is trapped in a single minimum withinsufficient kinetic energy to surmount any activationbarriers. A more phenomenological distinction is theappearance of a non-zero shear modulus on transitionto a glassy state.

1.3. Strong and fragile liquids

Glasses-forming liquids can broadly be categorised ona scale, for which the extrema are strong liquids andfragile liquids. Strong liquids tend to have opennetwork structures with strong directional bonds whichresist temperature-induced structural change. Fragileliquids, on the other hand, are characterised by lessdirectional interactions (for example, Ionic and Van derWaals) [32]. This delineation is not arbitrary; strongliquids such as SiO2 have small or undetectable changesin their heat capacity Cp in the temperature range overwhich the relaxation timescale crosses the experimentalthreshold. In such cases, distortion of the structureis restricted by strong directional bonds, constrainingthe Si atoms in SiO2 to be tetrahedrally coordinated,and resulting in a low density of configurational states.For a fragile liquid, the degeneracy associated with anincrease in energy is much higher, and the structure is“smeared” out over a continuous temperature range.Cp must be proportional to k lnW , where W is thedensity of states, hence the sharp increase in heatcapacity at the glass transition [33]. Essentially, afragile liquid samples more metastable structures.

Qualitatively, the two can be distinguished bymeans of an Angell plot (figure 1) of the logarithmof the viscosity against scaled temperature. The mostwidely used metric of fragility is the steepness indexm, or the slope of the Angell plot [34]:

m =d log〈η〉d(Tg/T )

∣∣∣∣T=Tg

. (1)

It has recently been shown that the fragility of aliquid can be predicted using the linear elasticityof the glass, ‘a purely local property of thefree energy landscape’ [35]. Quantitatively, thedistinction between strong and fragile liquids arisesin the temperature dependence of the single particlediffusivity D (which can be evaluated from moleculardynamics simulations) or equivalently the viscosityη. For strong glasses such as SiO2, η has anArrhenius form, suggesting well-defined energy barriers

5

log(

) (p

ois

e)

Tg/T0 1

-4

13

STRONG

FRAGILE

(o-terphenyl)

Figure 1. The Angell plot of viscosity (η) against scaledtemperature (adapted from reference [12]). Strong liquids tendto be close to a straight line, while fragile liquids are representedby lines that curve towards the bottom-right. Most liquids existin the shaded region, between the strongest (silica) and the mostfragile (o-terphenyl).

to diffusion, while weak liquids have a η that increasesat a greater, non-Arrhenius rate. This has beenattributed to a collective rearrangement of groupsof particles in a fragile liquid, compared with smalllocal rearrangements due to the strong directionalbonding in a strong liquid [36]. This fragile-strongbehaviour is thought to be intimately connected withthe phenomenon of polyamorphism in glasses [37]. Adynamical crossover temperature TB , well above Tg,was established as the strong-fragile transition for o-terphenyl recently [38], and it has also been suggestedthat a liquid-liquid phase transition may result in thechange in fragility [39].

How does fragility affect glass-forming ability?The two should not be conflated, since theirrelationship is not simple, and the advent of vapourdeposition methods for preparing ultra-stable glasseshas made it even more unclear. Fragile liquids arebetter glass formers when cooled from the liquid phase,and all molecular systems which have been shown toform stable glasses have intermediate-to-high fragilities(m), although it has been shown that organic moleculeswith low fragilities can form stable glasses via vapourdeposition [8].

1.4. The glass transition

According to Angell, the essential feature of the glasstransition is the appearance of a huge gap between thevibrational and structural equilibration timescales, of

up to 17 orders of magnitude [12], but it is difficultto associate this change with a thermodynamic phasetransition. The glass transition resembles a secondorder phase transition in the Ehrenfest sense, withcontinuous volume and entropy changes, but it cannotbe defined as a genuine phase transition since crossovertemperature is cooling rate-dependent [17]. It is notgenerally thought to be a thermodynamic first ordertransition, since no latent heat has been measured.In the case of metallic glasses, this question is stillopen, since the the latent heat may be small andtherefore difficult to measure, and there is evidencefor a first order transition from experiment [40] andsimulation [41]. Some simulations suggest that it maybe a different type of transition, a non-equilibriumfirst order transition in trajectory space, controlled byvariables that drive the system out of equilibrium [42].

The transition from a supercooled liquid to aglass is generally vaguely defined, and it can becharacterised by a number of different temperaturesor temperature regimes. There are three transitiontemperatures that commonly appear in the literature,the experimental glass transition temperature Tg, theKauzmann temperature TK (see section 1.5) andthe mode-coupling transition temperature TMCT (seesection 2.1).

The experimental glass transition temperature Tgis defined as the temperature at which the liquid, oncooling, exhibits a sharp decrease in heat capacity(Cp), to a value approaching that of the crystallinestate. There is no standardised value for the coolingrate, but Tg is usually defined as the temperatureat which the onset of the Cp increase occurs duringthe heating of a glass at a rate of 10 K min−1.Despite an apparent lack of consistency, Tg is easyto determine, and there is often good agreement, towithin 2 K between independent groups, when the onlyreproduced condition is the cooling rate [43]. Thistemperature is, however, insufficient to describe allaspects of the glass transition. The aforementionedtimescale difference can vary by as much as 10 ordersof magnitude between Tg and 1.1Tg [12]. The heatcapacity peak, and plateau in thermal conductivity aregenerally assumed to result from a universally observedanomaly in the vibrational density of states of glasses,identified by an excess of low energy states above theacoustic modes, called the “boson peak.” The bosonpeak has been linked to a transverse acoustic van Hovesingularity in the corresponding crystalline phase [44].

In the vicinity of Tg, the relaxation behaviourof a supercooled liquid diverges from the exponentialArrhenius form, instead conforming to a stretchedexponential of the form exp[−(t/τ)α] (often describedusing the empirical Vogel-Fulcher-Tamman, or VFTlaw; see section 1.5), where the stretching exponent

6

α lies between 0 and 1 [45]. It has been suggestedthat a discontinuity in the relaxation time at thecrossover from the Arrhenius to the VFT regime (asthe temperature decreases below Tg) might constitutethe aforementioned phase transition. Measurementson three good glass formers reveal a sharp crossoverbetween the two regimes with a width of less that 15 K,that cannot be accounted for by phenomenologicalmodels [46].

The definition of a laboratory glass transitionremains somewhat vague, since the properties of theglass formed vary with the cooling rate, with a fastercooling resulting in a higher Tg. An ideal glasstransition can be defined in the limit of an infinitecooling rate; above the ideal transition temperature,the behaviour of the system is ergodic, and it has timeto explore every possible metastable structure, untilit ends up in the most thermodynamically favourablenon-crystalline state at the global minimum state ofthe potential energy landscape [47]. The characterof this non-crystalline minimum, or ideal-glass stateremains unclear; for example, is there some local degreeof crystallinity, and if so, to what extent are crystallineinclusions allowed in an amorphous state [48]?

1.5. The Kauzmann paradox

Kauzmann defined an ideal glass transition tempera-ture, resulting from a paradox [30]. Using thermody-namic data from a variety of molecular glass-formers,he demonstrated that the excess entropy of the liquid(Sex) with respect to the entropy of the crystal (Scryst)vanishes at some temperature below Tg, named TK . Inother words, a graph of Sex/Scryst against temperaturereaches zero at a finite temperature [12] TK , at whichpoint the liquid and crystal have equal entropy, as il-lustrated in figure 2.

Decomposing the total entropy into vibrational(Svib) and configurational (Sconf) components, we have

S = Svib + Sconf , (2)

and we define the excess entropy as,

Sex = Sliq − Scryst (3)

It is commonly assumed that the vibrational contribu-tions are similar for liquid and crystalline phases, suchthat,

Sex ' Sliq,conf − Scryst,conf = ∆Sconf (4)

It seems counterintuitive that a liquid or amorphoussolid can have a lower configurational entropy than acrystal below TK , and this is the apparent paradox.However, there is no general principle constrainingthe entropy of a liquid to be greater than that of a

Absolu

te e

ntr

opy

Heat

capacit

y

T

T

Tg Tm

TK Tg Tm

0

0

crystal

liquid

stable

liquid

supercooledliquid

cryst

al

superc

ooled

liquid

Figure 2. Illustration of the Kauzmann paradox (adaptedfrom reference [50]), showing the increase in heat capacity asa liquid is cooled below its melting temperature Tm, and thecrossing of the absolute entropy curves for the liquid and thecrystal at the Kauzmann temperature TK . The shaded regionis experimentally inaccessible for a glass due to the slowing ofdynamics below Tg .

crystal [49]; indeed, Kauzmann notes that it is possibleto conceive a situation at lower temperatures, wheretighter binding of a molecules in a highly strainedliquid structure would result in high vibrationalfrequencies and a lower density of vibrational states.

It turns out that TK correlates impressively wellwith the divergence temperature T0 of the phenomeno-logical Vogel-Fulcher-Tammann (VFT) law. It intro-duces a different measure of fragility in the form of theparameter D, which quantifies the deviation from theArrhenius law [51],

η = η0 exp

(DT0

T − T0

), (5)

where η is the viscosity. The correlation is reflectedin the excellent agreement between Tg/TK and Tg/T0

[43], and the fact that TK ' T0 for a wide range ofmolecular and covalent glass formers [52]. However,this relationship is not universal, and breaks downfor many glass formers. Here, there is instead apositive correlation between TK/T0 and D [53]. Thisrelationship is attributed to short range ordering in thecase of strong liquids, modelled by the formation of theglass from superatomic rather than atomic units.

7

When modelling glasses, it should be noted thatneither the vibrational nor configurational entropy isproportional to the excess entropy (as demonstratedfor liquid Se), and that the anharmonic contributionto the vibrational entropy cannot be ignored [54, 55].

In practice, TK is an experimentally inaccessiblequantity because the glass transition intervenes beforethe excess entropy can reach zero, but its value can beinferred from the VFT equation. For the same reason,the Kauzmann temperature has not been directlyobserved in dynamical simulations, and its existencehas only been inferred from the extrapolation of hightemperature data [56].

The existence of the Kauzmann entropy crisisis supported by the Adam-Gibbs equation whichrelates the relaxation timescale to the configurationalcontribution to the excess configurational entropy Sconf

[36]:

τ ∝ exp

(C

TSconf(T )

)(6)

This expression is based on the concept of weaklyinteracting subsystems within the glass, expressingthe cooperative motion of distinct spatial regions inaddition to individual atoms. Although it has beenargued that the underlying arguments are weak, itdemonstrates a divergence in relaxation time as theexcess configurational entropy tends to zero, andhas also been shown to be in good agreement withexperiment.

The Kauzmann paradox can be viewed as evidencethat the glass transition is essentially a thermodynamicphenomenon masked by slow kinetics [12], and impliesthat it may be possible to predict glass forming abilityvia the potential energy landscape rather than the freeenergy landscape.

2. Models of the glasses and the glasstransition

In this section, we discuss the most importanttheoretical frameworks used to describe glasses and theglass transition. We mention mode-coupling theoryand random first-order transition theory only brieflythough, since they are beyond the scope of this review,and are discussed in detail elsewhere [57–60].

2.1. Mode-coupling theory (MCT)

The mode-coupling theory, developed independentlyby Leutheusser [61] and Bengtzelius et al. [62]characterises the dynamics of the glass transition usingthe kinetics of a simple hard sphere model.

Starting from the density of particles ρ in a liquid[59],

ρ(r, t) =∑i

δ(r − ri(t)), (7)

which has the Fourier transform,

ρk(t) =∑i

∫dreik·rδ(r − ri(t)) =

∑i

eik·r. (8)

The correlation function of interest is the dynamicstructure factor F (k, t):

F (k, t) =1

N〈ρ−k(0)ρk(t)〉 =

1

N

∑ij

〈e−ik·ri(0)eik·rj(t)〉.

(9)The MCT equation (derived in detail in reference [59])is the non-linear equation of motion of this time andwave vector dependent correlation function.

d2F (q, t)

dt2+q2kBT

mS(q)F (q, t)+

∫ t

0

dτK(q, t−τ)∂F (q, t)

∂τ= 0.

(10)Recasting this using Φ(t) ∼ F (k, t) neglecting thecoupling wave vectors results in the expression,

∂2Φ(t)

∂t2+ Ω2

0Φ(t) + λ

∫ t

0

dτΦ2(t− τ)∂Φ(τ)

∂τ= 0. (11)

At t = 0, F (k, t) is identical to the static structurefactor S(k) of a liquid, which can be measuredexperimentally. At high temperatures, above themelting point, Φ(t) decays exponentially with time,but for supercooled liquids, it undergoes a multi-step relaxation (figure 3). At short times, free andcollisional events result in the rapid decay in regionI, referred to as α-relaxation. The plateau of regionII is a result of the trapping of particles in cagesformed by their nearest neighbours, and the subsequentdecaying regime corresponds to cage-breaking events,is known as the β-relaxation regime. At long times, theβ-relaxation takes a stretched exponential form.Thesolution for the case of a glass is sketched in figure 3.MCT is characterised by a sharp transition froman ergodic regime to a non-ergodic regime as thetemperature is decreased below the mode-couplingtemperature TMCT. This behaviour is not systemdependent, and applies to both metallic and non-metallic glasses [25].

MCT makes some very detailed predictions thathave been experimentally verified (particularly withrespect to α and β-relaxation), and is generally validin the vicinity of the glass transition. The MCTsingularity is not equivalent to the laboratory glasstransition described in section 1, since it occurs 30-50 Khigher than Tg, and it appears to work considerablybetter for fragile liquids than for strong liquids [28].Moreover, its most obvious failing is the predictionof a sharp transition, contrary to what is observedin experiments. It has been successfully testedagainst molecular dynamics simulations for a variety ofsystems, including binary Lennard-Jones soft spheres

8

ln(t)

(t)

(t)

(t)

simple liquid

Figure 3. Wave vector-independent solutions to MCT equation(11) for three cases (adapted from reference [59]). A simpleliquid is characterised by exponential decay in the correlationfunction. A supercooled liquid has three distinct regimes:I. short timescale decay from free and collisional events; II.intermediate plateau, particles are trapped in cage comprisedof nearest neighbours (β-relaxation); III. long timescale non-Arrhenius (stretched exponential) decay. The decay in thetransitions to and from regime II (IIa and IIb) obey power lawswith respect to time. The glass has a singularity in the solutionof the MCT equation at TMCT; this results in the “freezing” ofthe structure in the non-ergodic regime.

[63, 64], binary hard spheres [65] and liquid silica [66].In the latter, MCT calculations are performed usingonly structure factor data from molecular dynamics,and while there are discrepancies (in particular forthe crossover temperature), a remarkable degree ofquantitative agreement is shown for this ab initiotheory. A more complete summary of the current stateof MCT is available in references [58, 59].

2.2. Random first-order transition theory (RFOT)

RFOT is characterised by two distinct transitiontemperatures: a dynamical transition correspondingto ergodicity-breaking which can be formally identifiedwith the mode-coupling temperature, and a secondtransition at a lower temperature corresponding toa thermodynamic phase transition to a glassy phasecharacterised by a discontinuity in an order parameter,but no latent heat [67]. When a liquid is cooled belowits glass-transition temperature, it can be consideredto undergo a phase transition analogous to the firstorder crystallisation transition. Particle motion is asuperposition of translations to a new position andvibrations about a mean position; above the transition,the motion is dominated by translations, and below it isdominated by vibrations, with only occasional moves.Below the glass transition temperature, the particleis essentially trapped in a cage formed by its nearestneighbours and is unable to move due to the highvacancy formation energy. Crystals have a periodicdensity profile ρ(r), but this can be generalised to thefollowing for non-periodic structures [68].

ρ(r) = ρ(r, ri) =∑i

(απ

)3/2

e−α(r−ri)2 (12)

Here, ri is the set of coordinates of a genericaperiodic lattice of average lattice spacing a anddensity n = 1/a3. α is a variational parameterdescribing the localisation of a particle in a cage formedof its neighbours, for which a completely delocalisedliquid has α = 0, and an (meta)stable structure hasα = α0. The quantity 1/

√α0 is the vibrational

amplitude of the metastable structure and is directlycomparable to the Lindemann length dL,

dL =1√α0

(13)

The Lindemann ratio dL/a is approximately 0.1 forcrystals, and roughly defines the point at which thevibrational motion becomes comparable to the nearest-neighbour distance, resulting in melting. It turns outto be universal, and only weakly dependent on thenature of the bonding [69]. A liquid-crystal transitionis first order, and similarly a liquid-glass transition isfirst order in α; however, in contrast with a singledistinct crystal structure, a glass can be considered tobe an ensemble of metastable structures with a randomdistribution of free energies (hence the name RFOT).This results in a multiplicity of distinct structuralstates with an infinite lifetime, thus there is no latentheat, but instead a heat capacity discontinuity.

RFOT is built on the density-functional theoryof freezing (a static treatment of mean-field theory),

9

and turns out to be formally equivalent to mode-coupling theory (a perturbative treatment of mean-field theory; see section 2.1) [70]. In the RFOT, glassystates correspond to the minima of the free energyF as a functional of inhomogeneous number densityn(x), corresponding to a non-uniform density field.Starting from a random close packed structure, a trialdensity field, containing a parameter that describesboth uniform and non-uniform states, is constructed.It is possible to calculate the free energy as a function ofthis parameter, such that it is possible to find the pointat which an non-periodic solution becomes possible.The free energy functional can be approximated to [71],

F [n(x)] = F0[n] + Fint[n] (14)

=

∫dxn(x)[lnn(x)− 1] (15)

− 1

2

∫dx

∫dx′n(x)C(x− x′)n(x′) + Funi,

(16)

in which F0 is the ideal gas free energy functional,Fint is the lowest order term in the expansion ofinteraction part of the free energy and Funi is aconstant term contributing to the free energy of theliquid. C(x) is the liquid-state direct correlationfunction, a renormalised form of the bare interactionpotential [69]; its Fourier transform C(q) is related tothe measurable static structure factor S(q) (the Fouriertransform of the radial distribution function) by

nC(q) = 1− 1

S(q)(17)

A dynamical RFOT can also be derived, and theresulting dynamical equation is,

δFsδns(x, t)

= kBT [lnns(x, t)−∫dx1C(x−x1)n(x1, t)+. . .]

(18)Glassy states are non-periodic solutions of

δF [n]

δn(x)= 0, (19)

An alternative description of the glassy state arisesnaturally from RFOT, the “mosaic scenario.” Mode-coupling theory is based on the concept of dynamicheterogeneity, and the glassy state is described bydynamical correlation functions; however, the mosaicscenario defines a static correlation length in anensemble of inequivalent states below the mode-coupling temperature [72]. Bouchaud and Biroli [67]construct a model starting with a non-periodic systemconsisting of a spherical region of Lennard-Jonesparticles of radius R in a metastable configurationα, with hard boundary conditions, i.e. the particles

surrounding the sphere are frozen. The free energy costof rearranging this configuration to another state, β,then consists of two opposing contributions: a surfacetension component due to the change at the boundary,and an entropic contribution from the change in theconfiguration. The glassy state can be described by thelength scale ξmos: if R < ξmos, then the configurationmay “flip” from α to β, but the pinning field of thefrozen boundary will force it to flip back. For R > ξmos,a change in state is thermodynamically favoured, buttakes a longer time due to the higher degeneracy, soξmos becomes the size of the cooperative region [72].Therefore, large clusters are thermodynamically in aliquid state, while a glass is formed of a “mosaic” ofmesoscopic, locally metastable domains, separated bystrained but mobile walls. The relaxation within amosaic domain is an activated process driven by theconfigurational entropy [51, 67]. These domains havebeen shown in simulations to change from sphere-likeat low temperatures to string-like or “fractal” at TMCT

[73].There is some controversy over the validity of the

mosaic scenario, with some studies suggesting a possi-bly contradictory one-state scenario [72, 74], and oth-ers producing evidence in its favour [75]. Interestingly,these domains have been directly observed [76] andtheir size is consistent with predictions, around 90–200 molecules [69, 73]. Bhattacharyya et al. have at-tempted to unify RFOT and MCT to describe the dy-namics at low temperatures as well as around TMCT,and have thus been able to predict the fragility parame-terD and the stretching exponent in the non-Arrheniusregime [77].

2.3. The potential energy landscape (PEL)

The static structure, or state of a system of N particlescan be described as a point in a 3N -dimensionalconfiguration space; its dynamics can then be viewedas as the motion of a state point r at an elevation ofpotential energy φ(r) above a 3N -dimensional energylandscape at zero temperature. This landscape ischaracterised by local minima for which dφ/dr =0; these generally represent metastable states of thesystem, and there exists a global minimum, whichis the thermodynamically stable state of the systemat absolute zero. The local minima, or basins, maybe separated by barriers which represent the energycost of breaking and making bonds to move fromone (meta)stable state to another. A one-dimensionalillustration is given in figure 4.

If we are not interested in the kinetics of thesystem, it is helpful to consider only the basins ofthe system. Stillinger et al. argued first empiricallyand later more rigorously that the number of minimaΩ(N) in an monatomic N -atom system follows the

10

Pote

nti

al en

erg

y

Collective con gurational coordinate

saddle

Figure 4. Schematic representation of the potential energylandscape (adapted from reference [78]). The horizontal axisrepresents the configurational degree of freedom, i.e. 3N spatialcoordinates. This landscape is more reminiscent of a glassformer, since a randomly chosen coordinate on the x-axis is morelikely to correspond to a metastable amorphous state.

relationship [79, 80],

Ω(N) ∝ N !eαN , (20)

where α is a positive constant in the large-N limit.In nature, configurations are found in their lowestenergy states because these have the largest Boltzmannweights in the canonical ensemble, which mitigates thetimescale required for ergodicity. It is obvious thateven for small finite systems, exhaustively sampling thebasins in configuration space via computer simulationis impossible, necessitating a limited sampling. How toachieve this is a question that has yet to be answeredfully, although some possible routes are suggested insection 2. Some small systems have been studied to theextent that their PEL is essentially well characterised,primarily in terms of the local minima. One suchexample is the thirteen-atom Morse potential (M13)cluster [81]. Such studies have elucidated generalfeatures of the PEL, for example the existence of“funnels” in systems with a tendency to crystallise:these are collections of mutually accessible deep basins,possibly around the global minimum, and act as trapsas the temperature is lowered.

How can one visualise such a high dimensionalspace? Qualitative schematic diagrams of one andtwo-dimensional surface are common as illustrations,but do not represent real PELs. Becker andKarplus introduced the use of disconnectivity graphsto map the partitioning of the PEL into local minimagrouped into superstructures [82] (see figure 6). Each

Figure 5. Heuer’s scheme for projecting the PEL for a modelsystem, a 32 atom binary Lennard-Jones glass (reprinted withpermission from Phys. Rev. Lett. 78, 4051 c©1997 AmericanPhysical Society [83]). Minima are grouped together along thehorizontal axis based on their maximum potential energy, andbarriers are indicated between minima. Note that adjacentminima on the diagram may be separated by large distancesin configuration space.

“leaf” represents a distinct local minimum, while thebranching structure describes how these local minimacoalesce to form basins of basins (metabasins), andhow these merge to form higher levels of structure.The effect of moving vertically up the disconnectivitygraph is equivalent to increasing the temperature. Asthe kinetic energy in the system becomes large relativeto the potential energy differences between the basins,the system does not “see” low energy basins with smallbarriers between them, but only the larger metabasin.If the temperature is raised enough, the system is freeto explore the whole configuration space, and only seesone superbasin.

Angell explicitly proposes a connection betweenthe topology of the PEL (namely the densityof configurational states) and the fragility of theassociated liquid [33, 43]. The PEL topology presentsthe resolution of the Kauzmann paradox: fragile liquidshave a PEL with a statistically insignificant number (oforder N) of deep minima, which contribute minimallyto the density of structural states, i.e. the probability offinding a configuration of a specific energy on the PEL.This low degeneracy results in a low configurationalentropy, comparable to that of a crystalline solid.Crystal structure prediction methods offer an insightinto the character of the PEL for crystal formers: itis expected to consist of a super-basin, or “funnel,”containing the global minimum crystalline state, whichitself occupies a disproportionately large hypervolumeof configuration space [84]. The strong liquid issomewhere in between, except the “ideal glass” stateis expected to occupy a region with a low density ofstates (i.e. occupying a large volume of phase space).In the case of both glass-forming liquids, the crystallineminima occupy statistically insignificant portions ofconfiguration space.

A comparison between the PELs of two cluster

11

systems at the extrema of the glass forming scale givesus some clues about the differences between crystal-forming systems and glass-forming systems. (KCl)32

tends to form rock salt-like crystals, and the Ar19

system, which is rare gas-like, tends to form amorphousphases in spite of the existence of a double-icosahedralstructure which is proposed to be the global minimum[85]. The (KCl)32 PEL can be described as “step-like,”whereas the Ar19 topology is “sawtooth-like.” (KCl)32

has a global minimum (rock salt) with basin energiesincreasing with distance from the global minimum,resulting in a funnel configuration. It crystallised evenunder rapid cooling of 1012 K s−1. In contrast, theenergies of the minima of the Ar19 system differedonly slightly apart from the global minimum; thissystem failed to find the global minimum even afterslow cooling at a rate of 109 K s−1.

The sawtooth/step dichotomy is a slightly simplis-tic scheme, but De et al. propose the distance-energy(DE) plot as a tool for distinguishing glassy clustersfrom non-glassy clusters on a similar basis [86]; theEuclidean root-mean-square distance (RMSD) in con-figuration space can be used as a “fingerprint” to deter-mine the similarity of two phases, and as a parameterto define a 1D PEL [87]. In particular, the density ofstates in configuration space near the global minimumis much higher for glassy systems: fragility seems tobe associated with a higher density of inherent states,lower activation barriers and higher vibrational fre-quencies [88]. A mathematical treatment of the densityof states demonstrates that there is a strong correlationbetween ground state degeneracy and the roughness ofthe PEL [89], which fits well with the general picturethat good glass-formers have a rough, highly degener-ate PEL with no easily accessible ground state.

Doye and Wales [90] draw similarities betweenthe potential energy surfaces and the free energysurface model used by Wolynes et al. to describeprotein folding [91, 92]. The protein folding studiesdefine temperatures Tf at which the “native structure”(global minimum) becomes thermodynamically stable,and Tg at which the protein is in a glassy state.A large ratio Tf/Tg then ensures that the systembecomes trapped in the native state rather than alocal minimum. They demonstrate that increasing thebarrier height in their simulations of Lennard-Jonesclusters corresponds to increasing Tg without changingTf , thus reducing the rate of relaxation to the globalminimum. Using this notation, Ar19 has a low ratioand (KCl)32 has a high ratio.

Abraham and Harrowell make the interestingobservation that inherent structures on the PEL ofamorphous solids have non-zero shear stresses [93].Considering the normal modes, they find that the onlywave vector with a non-zero shear stress is k = 0,

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25 .

.2.86

2.60

2.34

2.08

1.82

1.56

1.30

1.04

0.78

0.52

0.26

0.00

ΔE(LJunits)

ΔE(LJunits)

BLJ55

BLJ45

Figure 6. Disconnectivity graphs for the glassy binary Lennard-Jones 55-atom cluster and the crystalline 45-atom cluster(reprinted with permission from Phys. Rev. Lett. 112, 083401c©2014 American Physical Society [86]). The glassy cluster

has a multitude of near-degenerate states organised in manymetabasins, while the crystalline cluster has one large metabasinand a distinct global minimum which is much lower in energythan the next lowest state.

i.e. fluctuations that span the simulation cell; thiswould be eliminated if the shape of the cell was allowedto change. Thus we arrive at the seemingly trivialconclusion that the inherent shear stress results fromboundary conditions, and on a macroscopic level, theresidual stress will exist in any extended aperiodicsolid. One important conclusion to draw from this isthat the inherent structures all correspond to rigid solidstates since these must be able to support a stress bydefinition.

Doye et al. use disconnectivity graphs to map thePELs of various Lennard-Jones clusters, showing thatcertain cluster sizes have significantly different PELs(notably 31 atom cluster LJ31)), with a multitude ofmetastable states not much higher in energy than theglobal minimum, a sign of a good glass former [94].Disconnectivity graphs for 45 and 55 atom Lennard-Jones clusters are shown in figure 6 [86], demonstratingthe high degeneracy of states near the global minimumfor the glass former (BLJ 55). Heuer describes a

12

method of projecting a multidimensional configurationspace onto a one-dimensional potential [83], illustratedin figure 5. The energies of inherent structures werecomputed using extensive stochastic quenches (seesection 2), and the distances between these minima k inconfiguration space, calculated as a Euclidean distanced(k1, k2):

[d(k1, k2)]2 =

N∑i1=1

(ri1,k1 − r[i2(i1),k2])2 (21)

Here the difference in position of particle i1 betweenbasins k1 and k2 is computed, and summed overall particles. The notation i2(i1) in equation (21)expresses the fact that there there is no obviousmapping from particle i1 in basin k1 to the sameparticle in basin k2. Note that in a rigoroustreatment, particle mappings between configurationsand symmetries are taken into account. Reactionbarriers between the inherent structures are estimatedusing a quartic polynomial model. This informationis encoded in the transfer matrix V (k1, k2), whichcontains the saddle point energies for the path frombasin k1 to k2. It is then possible to construct a1D potential with an identical transfer matrix. Usingthis potential and a judicious grouping of the inherentstructures along the distance axis, a schematic of thePEL can be constructed. The model system in questionwas a 32 atom binary Lennard-Jones minimum, andthe numerical predictions extracted from the projectedPEL were accurate to within an order of magnitude.

2.4. The free energy landscape

In the potential energy landscape picture, we canindirectly recover the vibrational free energy of aninherent state by computing the local phonon densityof states; however, we neglect the configurationalentropy, which can be expected to have a largercontribution at low temperatures. If we want toconsider higher temperatures, we must look at thefree energy landscape, which includes an ensembleconfigurational entropy contribution at each statepoint, or configuration.

Although a purely thermodynamic description ofglasses with the PEL can get us a long way, it isnot a complete description at finite temperatures.This can be illustrated with the example of the 38-atom Lennard-Jones cluster (LJ38) [95]. It is well-characterised and has a double-funnel structure; theglobal minimum is a face-centred cubic truncatedoctahedron at the bottom of one funnel, and the next-lowest state is an incomplete Mackay icosahedron.Although the global minimum is thermodynamicallyfavoured, the incomplete icosahedron has a lower freeenergy at approximately two-thirds of the melting

temperature because the entropy is higher. Moreover,quenches from the liquid phase almost invariably endup in the icosahedron funnel since the icosahedronis closer to the polytetrahedral liquid structure. Inthe PEL picture, it can be said that the octahedronfunnel is narrow, whereas the icosahedron funnel isbroader, with a flatter basin. Close to melting, theicosahedral phase has a lower free energy. Uponentering the icosahedral funnel, the system will becometrapped there even when the free energy of the globalminimum phase is lower, indicating a high barrier.This is an example of a system that is pathologicallydifficult to globally optimise, and it gives us a greatdeal of insight into the differences between the PELsof crystals and glasses. Wales and Bogdan directlycompare the free energy landscapes of the LJ55 system,a “structure-seeker” (i.e. crystal former) with thefrustrated LJ38 system [96], from which it is possibleto establish a connection between glass-forming abilityand the efficiency of structure prediction algorithms.In addition to highlighting the double-funnel nature ofthe LJ38 system (figure 7), a feature on the free energylandscape points to a solid-solid phase transition fromthe octahedral global potential energy minimum to theicosahedral structure.

It seems, then, that glasses are governed bya complicated free energy landscape with manylocal minima from which the system cannot escape.Calculations on a hard sphere model suggest thatat a characteristic density (which is equivalent toa characteristic temperature in the context of thismodel), a large number of glassy minima appearin the free energy landscape [97]. Experiments oncolloidal glasses have provided the first direct evidencebacking up this assertion: since the waiting time of thestudied system in intermediate states varies, it can bededuced that there is a complex underlying free energylandscape, through which the system can take a varietyof routes to reach an equilibrium state [98].

The cooperative rearrangements described byAdam and Gibbs arises naturally in the free energylandscape picture. A rearrangement represents atransition from one basin to another, via a free energybarrier. Crucially, the path through the barrier inphase space can be described as a “simultaneouslyrearranged region”, or an excited state, which involvesmore atoms than in the equilibrium states [99]. Thisis visible in a glassy two-dimensional system of softdiscs [100, 101], and in a hard sphere model, in whicha “string-like” or looping movement of atoms appearsto provide a favourable path [102].

2.5. Discussion

We have described some different approaches to solvingthe problem of glasses and the glass transition that

13

replacemen

V

V

V

V

Q4

Q4

Q4

−279.2

−279.2

−279.2

−245.7

−245.7

−245.7

0.00

0.00

0.00

0.15

0.15

0.15

−254

−256

−258

−260

−262

−264

−266

−268

−270

−272

−274

−276

−278

−280

T = 0.30

T = 0.28

T = 0.26

lnGIS(V )

Cv/kB

kBT/ǫ

0.25 0.30 0.350.20

200

400

600

800

1000

1200

1400

Figure 7. Disconnectivity graph (where V is the potentialenergy), and heat capacity (Cv) of the LJ38 cluster (reprintedwith permission from J. Phys. Chem. B 110, 20765 c©2006American Chemical Society [96]). Insets show interesting partsof the free energy surface, as a function of potential energy andQ4, a bond-orientation order parameter. The horizontal axis ofthe disconnectivity graph ln(GIS) is the logarithm of the densityof states, which is directly related to the configurational entropy.At a low temperature (T = 0.26), there is a single free energyminimum, corresponding to the Mackay icosahedron, while atthe melting temperature temperature (T = 0.28), there is adouble funnel structure separating the liquid-like and solid-likestructures, and at a high temperature (T = 0.3) there is oneliquid-like structure.

have different starting points, but are nonethelessclosely related. RFOT is a framework that unifiesthree seemingly incongruous approaches: Adam-Gibbstheory, mode-coupling theory and spin-glass theory[49]. In particular, RFOT takes a similar approach tothe cooperatively rearranging regions of Adam-Gibbstheory, and MCT and RFOT have identical solutionsin the limit of an infinite dimensional system[70].

Although RFOT does not explicitly use the PELformalism in its derivation, the two are closely related,since in the formulation of RFOT a large number ofdistinct metastable states is assumed in the glass phase,a feature that arises from analysis of the PEL of glassysystems. Moreover, the dynamic transition (analogous

to the mode-coupling temperature) signifies the lossof ergodicity, as the system no longer has the kineticenergy to hop between the degenerate glassy states inthe PEL picture.

One might ask whether there is a place for aseemingly qualitative approach like the PEL whenthere are sophisticated first-principles approaches suchas MCT and RFOT available. In the words of Das [58],

‘. . . the competition of crystallization withformation of the glassy state has not beenstudied from such a microscopic approach.A unified description of the amorphous andcrystalline state is justified from the existingconservation laws, which hold irrespective ofthe nature of the thermodynamic phase.’

The “competition” between the crystalline and glassphases is one aspect that the PEL may be well suitedto describing, as discussed in section 8. The PELcan be mapped for many model and real systemsby calculating energies and interatomic forces usingclassical or quantum mechanical methods such as pairpotentials or density functional theory respectively.Statisical analysis of the PEL as discussed in section 5then allows a more quantitative analysis.

3. Theoretical background of the potentialenergy landscape framework

3.1. Thermodynamic formalism

Goldstein was the first to make the connection betweenthe PEL framework and the glass transition in asomewhat heuristic way, reasoning that the potentialbarriers separating basins dominate the dynamicalproperties at low temperatures (so-called β-processes)[103]. Events occur on two timescales; high frequencyvibrations within basins and Arrhenius-type activatedprocesses corresponding to jumps over barriers betweenminima on a long timescale. Stillinger and Weber [79]made the first steps in a thermodynamic descriptionof glass-formers using the topology of the PEL andthe concept of the inherent structure, a term wewill use interchangeably with “local minimum” and“basin”. Wallace expands on this, highlighting the roleof configurational entropy in the slowing of dynamicsin glasses using vibrational-transitional theory [104].This is also discussed in some depth in other reviewarticles [105, 106]. Working in the canonical (NVT)ensemble, with a system of N non-coincident particlesconfined to a volume V , a three-dimensional system has3N spatial coordinates r = r1 . . . rN. The potentialenergies of the set of configurations r, φ(r), definesthe PEL. It is important to note that the PEL is onlydependent on atomic coordinates, and is independentof temperature.

14

The crucial step is the discretisation of thecontinuous PEL to a set inherent structures. Relaxingthe atomic positions of an arbitrary configuration suchthat the forces are minimised using, for example, asteepest descent or conjugate gradient algorithm, givesthe local potential energy minimum in configurationspace. This allows the mapping of a continuous setof coordinates r surrounding a minimum to a singleminimum, α. This mapping M can be expressed asM(r) = α, and is equivalent to an infinitely fast quenchto zero temperature. R(α) is the set of configurationsthat quenches to α, and can be described as a basinin the PEL. These basins are separated by 3N − 1dimensional hypersurfaces for which the mapping Mis undefined.

The canonical partition function for a N -particlesystem, ZN , at temperature T is,

ZN =1

λ3NN !

∫e−βφ(r)dr (22)

where β = 1kBT

and λ is the mean thermal deBroglie wavelength. The configurational integral canbe separated into distinct quench regions or basinsR(α), so that

ZN =1

λ3NN !

∑α

∫R(α)

e−βφ(r)dr (23)

However, many inherent structures are identicalafter particle permutation, so basins belong to anequivalence class N !

σ . The symmetry number σ isunity for a system with hard boundaries, but the useof periodic boundary conditions imposes translationalsymmetry under which the potential is invariant, suchthat σ = N . Treating members of R(α) as equivalenceclasses,we have

ZN =1

λ3N

∑α

′ 1

σ(α)

∫R(α)

e−βφ(r)dr (24)

The ′ here denotes that any configuration with asignificant degree of crystallinity are excluded. Withinany basin, the potential energy can be recast as,

φ(r) = φα + ∆αφ(r), (25)

where φα is the basin minimum and the second termis the “elevation” from that minimum. Thus,

ZN =1

λ3N

∑α

′ 1

σ(α)e−βφα

∫R(α)

e−β∆αφ(r)dr (26)

The partition function is now separated into twodistinct factors: the discrete sum over e−βφα representsthe distinct temperature-independent configurations,while the integral over e−β∆αφ(r) represents thethermal excitations of the respective configurationswithin their basins.

3.2. Density of configurational states and entropy

To make use of this formalism, we need to express itin terms of parameters that can be extracted fromnumerical simulations. The density of states G(φ)counts the number of basins with energy φ, and can becomputed by sampling the potential energy landscapeusing the methods detailed in section 2. It expressedusing a delta function,

G(φ) =∑α

′ δ(φ− φα)

σ(α), (27)

and the total partition function can be evaluated bysumming over all basins:

ZN =∑α

G(φα)ZN,α (28)

From the thermodynamic connection, we can expressthe free energy per basin fα in terms of the partitionfunction per basin, ZN,α:

fα = φα − TS = −β lnZN,α (29)

Thus the total partition function is,

ZN =∑α

G(φα)e−βfα (30)

Defining the configurational entropy as Sconf =kB lnG(φ), the partition function becomes,

ZN =∑α

e−β(−TSconf+fα) (31)

As in section 3.1, the energy is essentially decomposedinto two components: one from the configurationalentropy, TSconf , and a free energy contribution fromvibrations within that basin, fα. The basins can nowbe considered as a continuum of levels characterisedby an energy φα and degeneracy G(φα). Whenthe inherent and vibrational states are in thermalequilibrium, the average configurational entropy canbe computed from [107],

dSconf(T )

dφα=

1

T(32)

Evaluation of the integration constant combined withan extrapolation to low temperatures would then inprinciple allow the Kauzmann temperature TK to becomputed as the crossover point of the entropies of theliquid and solid. This can be achieved by exploiting(29) to obtain the free energy of the liquid:

fliquid(T ) = −β lnZ(T ) (33)

= φα(T )− TSconf(φα)− f(β, φα) (34)

15

Here, φα(T ) is the mean inherent state energy attemperature T and the vibrational free energy ofthe basins at T , f(β, φα) can be approximated fromthe harmonic eigenfrequency spectrum of the systemcalculated using the basin distribution at T . Theconfigurational entropy can then be calculated asthe difference between the entropy of the liquid,via thermodynamic integration, and the entropy ofvibrational entropy of the solid as shown below.

3.3. Basin free energy

In condensed matter physics, it is common to assumethat when displaced slightly from their equilibriumpositions, the atoms in a crystal are subject to aquadratic potential — the harmonic approximation.The potential energy close to an inherent state can beexpanded quadratically [106],

V (r) ' φα +∑i,j,α,β

Hiαjβδrαi δr

βj , (35)

where H is the Hessian matrix, with elements,

Hiαjβ =∂2V (r)

∂rαi ∂rβj

, (36)

and δαi denotes a small displacement of atom i in direc-tion α. Solving for 3N eigenvalues, representing curva-tures, and 3N eigenvectors, representing displacementvectors, the system is equivalent to a set of 3N inde-pendent harmonic oscillators with frequency ωj . Thepartition function of a single basin can then be ex-pressed,

Zα = e−βφα3N∏j=1

1

β~ωj(φα)(37)

Averaging over all basins, this becomes,

ZN = e−βφα〈e−∑3Nj=1 ln[β~ωj(φα)]〉α (38)

Thus the basin free energy fα in the harmonicapproximation can be written,

−βfα = −βφα − βfvib (39)

= −βφα − ln〈e−∑3Nj=1 ln[β~ωj(φα)]〉α (40)

The vibrational free energy fvib is often approximatedto,

βfvib = −⟨

3N∑j=1

ln[β~ωj(φα)]

⟩α

(41)

Simulations on a binary Lennard-Jones model glass-forming system suggest that the harmonic approxima-tion is appropriate below the glass transition temper-ature, in the slow dynamics regime [108]. These re-sults also seem to indicate a transition from a harmonicregime below the crossover temperature to an anhar-monic regime above it, but it is unclear whether this istrue in general.

4. Schemes for sampling the PEL

One of the principal difficulties in modelling amor-phous solids is the construction of microscopic struc-tural models that are representative of the physicalreality. Amorphous materials are more difficult tocharacterise experimentally, since structural disordercauses broadening of spectral features that could beused to identify the atomic structure. In many cases,it is unclear whether a glass is disordered on the atomiclevel, or whether it has some form of superatomic unit,or shows signs of medium-range (quasicrystal-like) or-der. Our aims are twofold: firstly, we want to map outthe PEL such that we can determine whether the sys-tem is a glass- or crystal-former, and secondly, we needto develop physically representative models that can beused to perform computer experiments on specific ma-terials. In this section, we discuss several methods ofachieving these goals.

4.1. Reverse Monte Carlo (RMC)

The reverse Monte Carlo technique employs exper-imental data in the form of structure factors, theFourier transform of the radial distribution function,to create a consistent structural model. The pair dis-tribution function of a random configuration of experi-mental density is used to compute the structure factor,and the difference between the model and experimen-tal structure factors is used to define a generalised costfunction. A single atom is moved in the model configu-ration, and the move accepted or rejected according toa probability distribution determined by the cost func-tion. This algorithm will converge the model struc-ture factor to the experimental structure factor [109].RMC does not generate a unique structure, but a setof structures determined by constraints imposed by ex-perimental data. This may include structures with un-physical coordinations, but which are otherwise consis-tent with the experimental data. In extension to thismethod, experimentally constrained molecular relax-ation (ECMR), ab initio simulation methods are usedto obtain a model that is consistent between the ex-perimentally generated RMC data and the simulatedsystem [110].

4.2. Molecular dynamics: aging versus equilibrium

When a liquid is cooled, it falls out of equilibriumat the glass transition, i.e. the dynamics becomeso slow that it is impossible to equilibrate; beyondthis point the structure relaxes so slowly that it isextremely difficult to determine the functional formof the relaxation on an experimental timescale. Thisexperimentally imperceptible structural relaxation isknown as “aging.” Cathedral glass panes which are

16

thicker at the bottom are a common and most probablyapocryphal example of glass flow on long timescales;in fact, it is highly unlikely that such glasses age on ahuman timescale [111]. Fossilised amber, aged for 20million years presents a more convincing case for aging:it has a density 2.1% higher than thermally rejuvenatedamber, and provides evidence that the relaxation maytake a form that is not described well by the mostwidely used models, including VFT [112]. In fact,analysis of data for glass forming liquids has largelydiscredited VFT [113], and the best fit for relaxationin amber is the parabolic law model of Elmastad etal. [114, 115]. This not only brings into question thevalidity of the VFT law, but it also throws doubt ondependent theoretical paradigms such as the Adam-Gibbs formalism, and the existence of the Kauzmanntemperature, which arises naturally from the VFTform.

Molecular dynamics in principle allows thecomputation of time-dependent properties of a system,and should be the best method of modelling theliquid-glass transition. However, while it would beof great benefit to be able to model the evolution ofthe system in real time, given the structural disorder,it is prohibitively expensive to run simulations forlong enough to capture the slow dynamics of a glass.The time step must be chosen to be smaller than theshortest vibrational time period of the system, i.e.of the order of femtoseconds, which severely limitsthe potential length of simulations to millisecondsfor classical molecular dynamics and nanoseconds forab initio simulations. Longer timescales can beachieved using accelerated or coarse-grained moleculardynamics, but this also adds to the uncertainties.

Kob and Andersen [28] performed moleculardynamics simulations on a binary Lennard-Jonesmixture with a 4:1 ratio of atomic species and forcefieldparameters, both chosen to avoid crystallisation, the80:20 BLJ system. They observe that in manyglass-transition simulations, certain thermodynamicquantities such as the potential energy per particledisplay a discontinuity with respect to temperature,denoted Td. This simulation temperature is oftenmuch higher than the laboratory glass transitiontemperature Tg. In fact, Td is indicative of thesystem falling out of equilibrium, with an equilibrationtimescale greater than that of the simulation, i.e.a glass transition. Td depends on the cooling rateand the thermal history of the system. Moreover,monitoring the system via simple time-dependentvariables such as energy or pressure might suggest thatthe system is in equilibrium, as their values stabilise; amore careful examination using a two-point correlationfunction such as non-equilibrium generalisation of theincoherent scattering function Cq suggests an aging

process [116].

Cq(tw, tw + t) =1

N

⟨N∑i=1

eiq·[ri(tw+t)−ri(tw)]

⟩(42)

Here, tw is the time since the quench at t = 0, and qis a wave vector.

The van Hove self correlation function describesthe probability of finding a particle at distance r fromits original (t = 0) location at time t:

Gs(r, t) =1

N

N∑i=1

〈δ(|ri(t)− r(0)| − r)〉, (43)

and is essentially a time-dependent radial distributionfunction. Its spatial Fourier transform is the selfintermediate scattering function Fs(k, t) (identicalto the dynamical structure factor described insection 2.1). Using the same Lennard-Jones binarymixture Sastry et al. were able to associate theonset of the non-exponential relaxation characteristicof the glass transition with a well-defined temperaturewhich corresponds to the trapping of the system in thedeepest accessible basin on the PEL [117]. Definingthe relaxation time τ as the time when Fs(k, t) reachesa value of 1/e, they identify three distinct temperatureregimes: the first is the high temperature regime,in which the system has sufficient kinetic energy tosurmount activation barriers and explore configurationspace, and Fs has an exponential form. As thetemperature decreases to the “landscape-influenced”regime, a stretched exponential form Fs(k, t) =exp[(−t/τ(T ))β(T )] (where 0 < β < 1) becomesa better fit. In the “landscape-dominated”, whichcorresponds to just above TMCT, small secondarypeaks are observed in Gs, implying the occurrence ofrare jumps by particles over a distance of about theinterparticle separation. Thus at high temperatures,the system explores parts of the PEL with low minimabetween basins, and at temperatures close to TMCT,minima separated by considerably higher barriers areexplored. In the PEL picture, the dynamics becomedominated by the landscape at a crossover temperatureclose to TMCT [118].

Most studies have the explicit aim of examiningequilibrium properties, but one in particular usesnon-equilibrium molecular dynamics to describe theaging process in a glass [119], notably demonstratingusing the 80:20 BLJ system, that non-equilibriumdynamics are dominated by the connectivity of thebasins in configuration space, and by extension, theheight of the barriers separating them. The agingof the system is shown to be determined by atime-dependent effective temperature Te, defined byinverting the time-dependence of the temperature-dependent basin energy φα(T ) to get Te(φα). As

17

the system ages, it locates deeper basins, reducing itsvalue of Te, until it reaches the point when the barrierheights become of the order of kBT . At this pointthe exploration of configuration space changes frommoving along unstable modes to an activated processin which barriers must be surmounted. This activatedprocess is considerably less efficient for exploring thePEL. Aging can thus be considered the exploration ofprogressively deeper basins in the PEL picture; thisconcept has been extended by considering the shapeof the basins during aging, and its dependence onthe temperature to which the system is quenched.Specifically, for the 80/20 binary Lennard-Jonessystem, basins of a higher curvature (and thereforehigher vibrational frequency) are explored below theTMCT [120]. In the case of supercooled molecularliquids, it has been shown that rotations occur over alonger timescale than translations, and the time takenfor rotational correlations to reach equilibrium is morestrongly temperature-dependent than for translationalcorrelations [121]. Lempesis et al. introduce a variationon the familiar disconnectivity graph, the temporaldisconnectivity graph. A time (or equivalentlytemperature) axis is added to a disconnectivity graphto turn it into a 3D plot, and demonstrate the effectsof aging on inter-basin communication [122].

Tsalikis et al. also present results describing threedistinct temperature regimes, but these are of adifferent nature. High above Tg, the sampling ofthe PEL is ergodic, and is relatively computationallyinexpensive. In the second regime, in the vicinityof Tg, the sampling is inefficient, and in the third,below Tg, the system is essentially trapped indefinitely[123]. The most interesting conclusion of this workis that the nature of the glass formed is stronglydependent on the time spent by the system in thesecond temperature regime. They conclude that thecooling rate, particularly when it is conducted ina stepwise manner as in most molecular dynamicssimulations, is not sufficient to define the resultingglass, since the time spent at each temperature hasan extremely non-linear effect.

La Nave et al. used instantaneous normal modeanalysis (see section 5.3) to estimate the fraction ofdiffusive directions fdiff of equilibrated representativeconfigurations of silica at temperatures both aboveand below the crossover [124]. A direction on thePEL is considered to be diffusive if, when displacedin that direction, the system moves into a differentbasin, and fdiff is defined as the ensemble average〈Nu/3N〉, where Nu is the number of directions withnegative eigenmodes excluding those displaying signsof anharmonicity. It quantifies the amount of time thesystem spends on the ridges of the PEL between basins.fdiff is seen to decrease as the temperature decreases,

and vanishes at TMCT, implying that the number ofdirections leading to a different basin decrease withtemperature, which makes sense given that below TK ,the system is trapped in a single basin. Moreover,for silica, they find the direct relationship Sconf ∝log fdiff between the configurational entropy Sconf ,which essentially enumerates the number of distinctbasins explored by the system, and fdiff . This seemsto be system-independent, and holds for both strongand fragile dynamics.

Similar studies lead to some even strongerconclusions. Saddle points on the PEL are classifiedaccording to the number of imaginary modes inthe Hessian matrix, ns(T ), and the energy of thatconfiguration, es(T ). At TMCT, ns vanishes, and athigher temperatures it increases with a well definedfunctional form. This makes ns a good choice ofparameter for describing the dynamics of supercooledliquids, and it can be used to calculate TMCT withoutusing mode-coupling theory. In addition, the resultssuggest that above TMCT, the dynamics are entropydriven rather than activated, with the system followinga path through saddle points with a progressivelydecreasing value of ns [125, 126]. This findingsupports the validity of the “potential energy landscapeensemble” paradigm [127, 128] (see section 5.5). Sincethe Hessians are known at all saddle points, it ispossible to compute the partition function at thesaddles, which can be used in transition state theory[126].

“Melt-quench” molecular dynamics is the tra-ditional method of generating structural models ofglasses [129–131]. A crystalline phase simulated us-ing molecular dynamics is heated to well over its melt-ing temperature, and the resulting liquid is equili-brated. The temperature is then gradually loweredin a stepwise manner, equilibrating at each temper-ature increment, until the glass transition is reachedand the structure will no longer equilibrate on a rea-sonable timescale. However, the ability of this methodto generate representative amorphous structures is de-batable. In principle, the instantaneous structure of aliquid is indistinguishable from that of a glass; how-ever, the use of a single trajectory to generate manyquenched configurations may introduce unwanted cor-relations. Moreover, glasses are formed at low cool-ing rates in practice, and the simulation timescalesrequired for such low cooling rates are impracticalfor both classical and ab initio molecular dynamics.Whilst it is possible to simulate a low cooling rate usingsmall temperature drops followed by periods of equi-libration, the cooling rate is still orders of magnitudehigher than in practice, and it is unclear whether theresulting configurations represent those found experi-mentally. A further caveat is that this method is only

18

useful if the local structure of the liquid phase is sim-ilar to that of the amorphous solid phase; this is truein the case of SiO2 and GeSe2, but does not hold foramorphous silicon, which is tetrahedrally coordinatedas an amorphous solid, but sixfold coordinated as aliquid [132]. This discrepancy does not appear in thestructure factor of amorphous Si generated by melt-quench, which tends to contain the wrong density ofcoordination defects regardless of the level of theoryused in the potential [133]. This serves as a cautionagainst careless application of melt-quench moleculardynamics.

4.3. Density of states Monte Carlo

Monte Carlo sampling allows a time-independentsampling of the PEL. This has the advantage ofsidestepping the slow dynamics and equilibrationissues. Yan et al. perform Monte Carlo simulationsin the “multimicrocanonical ensemble,” in which theparticle number and volume are constant, but thetotal energy is allowed to fluctuate over a range[134]. A random particle is displaced slightly, andthe move is accepted with a probability determinedby the difference in energy between the states, asper the standard Metropolis Monte Carlo algorithm.Since the density of states is not known a priori,it is computed iteratively. The heat capacity canthen be computed from the density of states. Theresults of simulations on a 50:50 binary Lennard-Jones mixture show the peak in heat capacity andsharp drop in inherent structure energy as a functionof temperature characteristic of a glass transition.Crucially, the results conflict with direct moleculardynamics simulations [56] which used a power lawto extrapolate the entropy of the liquid to lowtemperatures and determine the intersection with theentropy of the disordered solid. The Monte Carlosimulations suggests that the power law does nothold at low temperatures, and that there is no suchcrossover (Kauzmann) temperature.

4.4. Continuous Random Networks (CRN)

Continuous random networks were originally conceivedto generate structural models for amorphous Si andGe, by preserving local order, while randomisingbond lengths and angles. Starting from a supercellof a periodic diamond structure, bonds are locallyrearranged by “rewiring” bonds which are modelledwith a naıve “spring” potential, maintaining theaverage coordination, but introducing five- and seven-rings in addiction to the six- and eight-rings in thecrystalline structure. The structure is then allowedto relax to within the constraints of the new nearest-neighbour list. This is repeated until the structure is

deemed to be sufficiently random [135].Variations on this technique involve starting

from a crystalline melt or a random four-coordinatedconfiguration, and using a Metropolis-type algorithmto accept or reject moves [136]. This method hasbeen applied successfully in modelling, for example,amorphous silicon [137] and silica [138], but itsapplication is limited to strong liquids, i.e. glass-forming materials where the atoms have a very definitecoordination number and form distinct covalent bonds.

4.5. Stochastic Quenching (SQ)

The SQ method bears some similarity to melt-quench molecular dynamics, but instead of rapidlyquenching snapshot configurations from a potentiallyexpensive and time consuming equilibrium moleculardynamics simulation of the respective liquid phase,a “sensible” random initial configuration is quenchedinfinitely fast. ‘Constrained stochastic’ configurationsare generated by randomly distributing atoms ina cubic box with periodic boundary conditions,subject to the constraint that no two atoms maybe too close, since this may interfere with electronicstructure optimisation, and cause instability ingeometry optimisation algorithms. Each configurationis optimized to the potential energy local minimum[139]. In materials where the characteristic lengthscale is unknown, a similar process can be performed,randomly placing superatomic units in the unit cellinstead of atoms; this may be appropriate whenthe crystal structure consists of easily identifiablesuperatomic units.

SQ is thought to be valid because it has beendemonstrated that, for a monatomic liquid, the vastmajority of all structures are random, and the normalmode frequencies are essentially the same for allrandom structures. The distribution of frequenciestakes the form of a Gaussian distribution with avariance that is independent of temperature [104, 140].Thus any randomly generated structure should inprinciple have the same structural properties.

SQ has been shown to accurately reproducestructural and chemical features of even complexamorphous systems such as ZrSiC. Disagreementsbetween theory and experiment have been attributedto the fact that SQ is effectively a quench withan infinitely fast cooling rate [141]. However, thisagreement may be a result of insufficient analyticalrigour. In the case of GeSe4 glasses, the structurefactors for stochastic quenched structures are verysimilar to those for structures formed from a newmelt quench technique in which the Ge atoms arelocalised in either a sphere or a slab embedded inamorphous Se. The NMR spectra tell a different story:when compared with experimental NMR spectra,

19

stochastic quenching leads to an overly homogeneousstructure, suggesting that the glass may be comprisedof superatomic units that only form during a melt [142].If it is possible to identify these superatomic units, itmay be more accurate to perform stochastic quenchingby randomly placing them in the cell rather than singleatoms.

This method is very similar to ab initio randomstructure searching (AIRSS) [84, 143], in whichensembles of sensible stochastic configurations inconsiderably smaller cells are quenched at zerotemperature to determine crystal structures. The keydifference in AIRSS is the exploration of the enthalpylandscape rather than the potential energy landscapeas in the SQ method.

4.6. Activation-Relaxation Technique (ART)

ART [144, 145] employs a completely differentmethod of exploring the PEL by taking into accountthe activation barriers between basins. Moleculardynamics simulations are incapable of sampling theconfiguration space since significant structural changeshappen on a timescale that is considerably longer thanexperiments, let alone simulation. In a glass, thesystem is essentially confined to vibrational motionwithin a basin.

Starting from a quenched configuration at a basinminimum, a single atom is displaced slightly such thatthe total force is non-zero. The configuration is thenpushed to a saddle point by following the PEL in adirection of increasing force (essentially the inverse ofa geometry optimisation); this may involve the motionof any number of atoms. Rather than using forcescomputed from the full dynamical Hessian matrix asin a phonon calculation, a “redefined” force G is used.

G = F −[1− α

1 + ∆x

](F ·∆x)∆x (44)

Here, F is a 3N -dimensional force vector derivedfrom the gradient of the potential energy, ∆x isdisplacement vector of the single atom from the localminimum and α is a parameter determining the rateof ascent to the saddle point.

From the saddle point, the system is quenchedinto a different local minimum. The net result isthe system hops from one basin to another. Thisprocess is repeated to explore the potential energylandscape. This method has the advantage thatits efforts are concentrated in the tiny region ofconfiguration space that is accessible (since mostof configuration space is comprised of unphysicalstructures); on the other hand, it could be arguedthat a bias is introduced by limiting the volumeof configuration space explored in this way. ARTis good at locating transitions between metabasins

(funnels) compared with molecular dynamics, in whichtransitions are weighted heavily by a Boltzmann factor[146].

4.7. Discussion

There are several methods for generating structuralmodels of glasses, but all have advantages anddisadvantages. Melt-quench molecular dynamics isby far the most widely used in spite of the factthat there are many reasons not to blindly trust it.In the cases where the liquid and crystalline phaseshave different first-neighbour coordination numbers,for example amorphous silicon, it fails miserably [133].The fact that there is a time-dependence betweenconfigurations may introduce unwanted correlations,and it is impossible to reach cooling rates closeto experiments due to the compuatational expense.Moreover, it is only physically representative of onemethod of producing glasses (cooling a liquid), and notvapour deposition, which is a highly non-equilibriumprocess.

The CRN method intriguingly works well situa-tions in which melt-quench molecular dynamics invari-ably fails (amorphous Si). This leads to the suggestionthat bond rewiring may be representative of the under-lying physics — indeed, it appears that bond switchingevents actually happen frequently over long timescales[145]. On the other hand, CRN is clearly inappropriatefor fragile glass-formers such as bulk metallic glasses,where there are no strong directional bonds.

When compared with stochastic quenching, melt-quench molecular dynamics consistently generateslower energy configurations; however, unlike in crystalstructure prediction, we are not searching for thethermodynamic ground state. In fact, experimentscomparing a quenched glass with a ultrastable vapour-deposited glass for the metallic glass Zr65Cu27.5Al7.5

demonstrate that the ultrastable glass has a Tg that is11 K higher, but has a considerably higher enthalpy [9].This is a point that is rarely considered and remainsan open question; similarly, when using stochasticquenching, it raises the question of how to choose themost representative structure from an ensemble.

5. Theoretical characterisation of glassystructures

In this section, we shift the emphasis from generatingstructural models of glasses to analysing them. Sincethese amorphous configurations lack the order ofcrystals, we need rigorous statistical methods fordescribing their structure and dynamics.

20

5.1. Structure and packing

It is commonly argued in the literature that it isimpossible to describe the glass transition using onlystatic structural models, and that it is necessary todescribe the dynamic heterogeneity of the system,i.e. the inhomogeneous distribution of frozen andcooperatively mobile regions as in RFOT (section 2.2).Instead of asking which aspects of the structure giverise to the dynamic heterogeneity, a question that hasyet to be answered, Widmer-Cooper et al. ask whichaspects of the dynamic heterogeneity arise from thestructure [147]. In a two-dimensional system of softdiscs, they find that while the initial configuration ofa dynamical system does not directly determine thelocations of diffusive regions, it does determine thepropensity of particles to undergo large displacements,where the propensity is defined as the ensemble averageof mean-squared displacements of a particle.

Traditionally, microscopic models of amorphoussolids have been based on random arrangements ofhard-sphere atoms, the dense random-packed model.However, the microscopic structure of a glass locallyresembles the crystalline state of the same composition,with a non-periodic packing of the same polyhedralbuilding blocks [148]. Attempts to describe liquids asdisordered crystals and dense gases have both failed,suggesting that a liquid is a well-defined phase. Thisis evidenced by the fact that simple liquids can besupercooled well below their melting point withoutcrystallising proving the existence of interphase energyin the form of the Gibbs free energy of crystallisation,and therefore the difference in liquid and solid structure[149].

A notable feature of metallic glasses is that theyhave a high density relative to the crystalline phase,up to 99.5% of the crystalline density in the caseof the best glass forming systems. This highlightsthe importance of efficient atomic packing, whichcan be best achieved with efficiently packing solute-centred clusters such as tetrahedra and octahedra. Therelative sizes of the constituent atoms are critical inthis regard [150]. Two ranges of order are presentin this scenario: short range order, namely orderingin the first coordination shell of the particles, andmedium-range order, the ordering between the notionalclusters resulting from the short range ordering. Sucha paradigm requires efficient space-filling on two levels,and it has been shown that icosahedral medium-rangeordering is a general feature regardless of the type ofshort range ordering present [151].

Frank was the first to suggest that atoms ina liquid cluster in the form of an atom-centreddeformable icosahedra [152]. It is impossible to fillspace with regular tetrahedra, so any attempt tofill space must involve distorted tetrahedra which

introduces strain into the system. This accountsfor the larger strain energies in liquids. Consideringclusters relaxes the space-filling criterion, so smallerclusters favour liquid-like polyhedral packings [153].The most favourable polytetrahedral packing consistsof icosahedra, which enhances the short range packing(in contrast with the face-centred cubic and hexagonalclose packed crystal systems, which improve the longrange packing) [149]. This goes some way towardsexplaining the resistance of supercooled liquids tocrystallisation — it requires a substantial atomicrearrangement. Moreover, five-fold symmetry isforbidden in crystals and is a barrier to crystallisation,but has been observed, for example, in simplemonatomic liquid lead [154]. Icosahedral ordering hasalso been demonstrated from a model of binary alloy,amorphous Ni-Ag, constructed using reverse MonteCarlo [155]. Experimental studies have also reportedicosahedral clustering in quaternary bulk metallicglasses, where the addition of a fourth species greatlyincreases glass-forming ability — in such systems, theratio of atomic sizes between solute and solvent isthe most important factor [156]. Whilst there isa broad spectrum of polyhedra present in metallicglasses, the dominant species varies with the atomicsize ratio [151]. There is evidence that the proportionof icosahedral structures in a glass is an indicator ofglass-forming ability. For example, lower cooling ratesresult in configurations containing more full icosahedra[157], and the glass-forming alloy Cu45Zr45Ag10 hasa higher proportion of icosahedron-like clusters [158].However, the characteristics of such glasses are not onlydependent on topology, but are also highly dependenton the chemistry of the constituent elements. Althoughdensity changes in amorphous solids experimentallycorrelate with glass forming ability, existing densepacking models have failed to predict the glass formingcompositions of the CuZr system, for example [159].

Based on X-ray and neutron scattering experi-ments, Ma et al. propose a fractal packing schemefor metallic glasses [160]. The polyhedral units of aquasicrystal cannot tessellate to fill space due to sym-metries (such as five-fold) forbidden in crystalline lat-tices, but instead tessellate to form a structure withouttranslational symmetry, and are fractals with a dimen-sion of 2.72. Similarly, the icosahedral solute-centredclusters that form metallic glasses cannot develop anyorder beyond a few cluster lengths, i.e. medium-rangeorder. This results in a fractal network with a dimen-sionality of 2.31, lower than that of a quasicrystal sincespace-filling is more difficult in a glass.

The solute-centred quasi-equivalent cluster modelis distinct from the simpler solute-centred modeldetailed above in that there is no orientational orderbetween the clusters, so that the solvent atoms occupy

21

random positions [161]. It is capable of predicting thenumber of solute atoms in the first coordination shellof a particular solvent atom, allowing the calculation ofatomic concentrations in bulk metallic glasses. Thereis also recent experimental evidence to support thismodel [162].

These solute centred clusters are often definedusing Voronoi tessellation analysis [163, 164]. TheVoronoi cell or polyhedron of an atom contains the setof points that are closer to that atom than any other,and is analogous to the Wigner-Seitz cell in crystallinestructures. The coordination of the central atomis described by the Voronoi index 〈n3, n4, n5, n6, . . .〉,where ni is the number of i-edged polyhedra and

∑i ni

is the coordination number [2]. Voronoi tessellationanalysis provides a rigorous definition of the freevolume of an atom, i.e. the space into which anatom is capable of moving, namely the volume ofits Voronoi polyhedron. By comparing the volumesof Voronoi polyhedra between liquids, glasses andcrystals, it has been shown that it is not possibleto distinguish a static snapshot of a crystal from aliquid or amorphous solid using distributions of freevolume [165, 166]. Modelling a 2D system of softdiscs, Widmer-Cooper and Harrowell conclude that themobility of a particle as characterised by its Debye-Waller factor (essentially the mean-squared deviationof a particle from its equilibrium position, averagedover all particles) is not dependent on its geometricalfree volume, since particles with a high free volumeare uncorrelated with particles with a high Debye-Waller factor — free volume is a strictly single-particleproperty, and as such fails to capture the cooperativedynamics over even short timescales. [167].

Free volume models are an attractive idea,supporting the notion that the critical point in theglass transition is when the structure changes tosuch a degree that there is no longer enough freevolume for atoms or molecules to percolate through theamorphous matrix. Free volume is just one elementof the description of percolation, but in order todistinguish between phases, one also needs to takecollective motion into account. It now seems that pastsuccesses of this model [168] may have resulted froman inconsistent definition of free volume.

5.2. Density of states

The configurational density of states (DOS) G(EIS)enumerates the frequency of local minima (inherentstates) of energy EIS. The canonical partition functioncan be written as a function of the density of states,G(EIS) [169]:

Z(T ) =∑IS

G(EIS) exp

(−〈FIS〉kBT

)(45)

and G(E) can be written as a function of theconfigurational entropy Sconf(E) [29]:

G(EIS) = exp

(Sconf(EIS)

kB

), (46)

and it should be noted that this excludes thevibrational contribution from all minima. The densityof states can be constructed by sampling the PEL andconstructing a histogram of inherent state energies.G(EIS) is commonly approximated as a Gaussianprobability distribution, according to the randomenergy model [169–171]:

P (EIS) =exp(αN)√

2πσ2exp

(−E

2IS

2σ2

)δE, (47)

where P (EIS) is the probability of finding an inherentstructure in the range EIS → EIS + δE, exp(αN) is anestimate of the total number of basins as in equation(20), and σ is the Gaussian width. This approximationdoes not hold for low energies, since the density ofstates of a glass former is characterised by a fat tail,such that there is a non-trivial probability of findinga structure with an energy close to the crystallineground state. This feature is not limited to amorphoussystems, however [172], but it may be possible todistinguish a glass-forming PEL from a crystal formingPEL from the functional form of the low-energy tail.

A comparison between a fragile liquid, a binaryLennard-Jones mixture, and a strong liquid, SiO2,shows a clear distinction in their respective DOS: SiO2

has a well defined cutoff energy, below which thedensity of states suddenly falls to zero, while the BLJsystem has a smooth G(E) that falls to zero as aGaussian [173].

Whilst it is impossible to directly sample thetotal number of local minima in a configuration space,a recent study on hard spheres has demonstratedthat it is possible to estimate it by calculating thehypervolumes of basin minima [174]. Given the basinvolume, it is possible to calculate the entropy of thedistinct minima, and thus the number of distinctstates.

5.3. Normal mode analysis

The application of normal mode analysis to liquidsand glasses allows access to a wealth of thermody-namic information, including the barrier-crossing rateand hence the diffusion constants. Most significantly,it presents the possibility of describing the dynamicheterogeneity of glasses from a static picture. Simu-lations on a 2D system of soft discs suggest that thelocations of regions of irreversible structural reorgani-sation (i.e. diffusion) are strongly correlated with the

22

positions of the local and non-local low frequency nor-mal modes of the initial configuration. Furthermore,positions of these low frequency modes persists overchanges in inherent structure [175]. These diffusive re-gions may correspond to the dynamic heterogeneitiesthat are thought to be necessary to describe glasses inrandom first-order transition theory [22].

Stillinger and Weber used local minima to describethe topology of the PEL, essentially ignoring thesaddles that separate them. One can also describethe PEL in terms of the density of saddles. Thecalculation of the normal modes of a static snapshotof a dynamic trajectory is known as instantaneousnormal mode analysis (INM). In contrast to a crystalstructure, liquids and glasses possess both stable(real) and unstable (imaginary) modes, such that thetotal density of vibrational states can be decomposedinto a sum of stable (ρs) and unstable (ρu) modes[176]. These unstable modes are thought to betransition pathways between basins. Zwanzig proposeda relationship between the basin-hopping frequencyωh and self-diffusion constant D(T ) [177] based onthe PEL formalism, and Keyes related the hoppingfrequency to quantities measurable in numericalsimulations, namely normal mode frequencies, and thesubset of normal mode frequencies that are unstable[178]. We define the frequency of a one-dimensionalconfiguration escaping a minimum of frequency ωm viaa barrier of frequency ω as ωe:

ωe = f(ω, ωm)e−βEA (48)

where f(ω, ωm) is a rate constant. Generalising this toan escape from a minimum on a 3N -dimensional PEL,we need to take into account the cross-section of the3N −1-dimensional barrier v(ω) and the volume of theminimum vm.

ωe =

(ω

ωm

)(v(ω)

vm

)f(ω, ωm)e−βEA (49)

Then the hopping frequency ωh is the integral of thisexpression over all minima and saddles.

ωh =

∫dω

(ω

ωm

)(v(ω)

vm

)f(ω, ωm)sn(ω)〈e−βEA〉ω

(50)The sn(ω) term integrates to the number of saddless connecting the minima. Using assumptions similarto the Adam-Gibbs law, the density of unstablevibrational states ρu can be expressed,

〈ρu(ω;T )〉 =( α

3z

)S(ω;T )

[1 +

∫dωS(ω;T )

]−1

(51)

S(ω;T ) =( sm

)(v(ω)

vm

)n(ω)〈e−βEA〉ω (52)

Here α/3z is the fraction of the 3z directions withdownward curvature corresponding to an unstablemode. The similarity in pre-factors suggests arelationship between ωh and the ρu, which can beevaluated by fitting a numerically computed density ofunstable states to a functional form; the troublesomesaddle point terms are also eliminated in this way.However, it should be noted that there may beimaginary modes in a crystal at finite temperature,i.e. non-diffusive unstable modes, and by extensionthese exist in liquids and glasses. The problem withINM is determining which of these unstable modes arediffusive [179].

5.4. Activated dynamics

Aging is a phenomenon that occurs on a timescaleorders of magnitude beyond what is accessible frommolecular dynamics simulation; diffusive processesthat occur beyond experimental timescales allow aglassy system to find lower thermodynamics states. Inthe PEL paradigm, diffusion in a supercooled liquidcan be described in terms of cage-breaking (i.e. aparticle breaking out of a cage formed of its nearestneighbours). Productive, or non-reversed, cage-breakscorrespond to transitions between metabasins, whereasnon-cage-breaking process correspond to non-diffusiveprocesses occurring within a metabasin [180]. Belowthe glass transition, the former are strongly suppressed,resulting in super-Arrhenius relaxation. This issue canbe circumvented by considering trajectories throughconfiguration space as discrete transitions betweenlocal minima. The problem of dynamics on the PELcan be greatly simplified by considering a transitionbetween two connected local minima as a two statetransition problem. When there is an energy barrierof height EA (an activated process), the statistics oftransitions between microstates can be modelled using(for example) an Arrhenius-type rate equation, wherethe transition rate is proportional to e−EAkT [181].

We now turn to how this enables dynamicalsimulations to be coarse grained and thereforeperformed on larger systems. Instead of using bruteforce molecular dynamics, the slow dynamics of a glasscan be modelled as a Poisson process of elementarytransitions between local minima, controlled by a rateparameter λ quantifying the expected number of eventsper unit time [123]. Such a (Markovian) process has no“memory” of the previous states of the system, muchlike radioactive decay. An “event” is defined as thetransition of the configuration from one basin to aneighbouring basin of higher energy. The cumulativeprobability P (t) is of the form,

P (t) =

1− e−λ(t−t0) t > t0

0 t ≤ t0(53)

23

where t0 is the time of the last event. Then the gradientλ is,

λ =n− rl∑n

k=r1+1(tk − tl), (54)

where n is the total number of transitions, tk isthe residence time of transition k and rl is the firstdata point to consider when calculating the slope(since the Poisson approximation does not hold atshort times). Tsalikis et al. [182] validated thismethod by comparing the results of the Poissonprocess with the mean squared deviation for themolecular dynamics simulations used to determinethe rate constant. This approach recovers the two-timescale behaviour that typifies glassy systems: shorttimescale transitions occur within metabasins, andlong timescale transitions occur between metabasins.Moreover, these transitions can be related withthe separation of the local minima in configurationspace as parameterised by the Euclidean norm:intrametabasin and intermetabasin transitions havedifferent characteristic “lengths”. These long distancetransitions correspond to collective rearrangements,whereas short distance transitions correspond toisolated cage-breaking events. It has been proposedto use this methodology to identify metabasins “on thefly” during parallel microcanonical molecular dynamicssimulations [183]. This model has been superimposedonto a network model (see section 5.5) of the PEL inorder to simulate aging dynamics [184]. A combinationof metadynamics [185] with the Markovian networkmodel has been adapted to compute the viscosity ofglasses, a feat which was previously impossible due tothe long-timescale dynamics [186].

A related approach employs continuous timerandom walks (CTRW), which are differentiated fromrandom walks by characterising each step by a variablewaiting time τ , during which the walker is stationary,followed by an instantaneous displacement ∆x. It hasbeen shown that such a scheme can be numericallyderived for a glass-forming system [187]. The CTRWscheme requires coarse-graining in order to apply itto the analysis of molecular dynamics simulationsfor two reasons: firstly, the particles are constantlymoving, rather than being stationary before each“jump;” thus it is necessary to carefully define whatconstitutes a jump. Secondly, a jump occurs over afinite time, rather than instantaneously as assumedby the model. Warren and Rottler demonstrated theviability of CTRW as a variation of the “cage” or“trap” model by parameterising it using ultra-longterm MD simulations of a binary Lennard-Jones glass.They observe that the key feature is that trap statesare uncorrelated, and the trap energy is chosen froman exponential distribution at each step [188, 189].Helfferich et al. detail the implementation of such a

scheme, adding the further constraints that particlesmay not loop back to their starting positions, oroscillate between two states [190, 191]. It should benoted that this coarse graining makes CTRW capableof describing the long-timescale β-relaxation (in theparlance of MCT) of the system, when a particlebreaks out of its cage, but not the short-timescale α-relaxations.

5.5. Network topology

The PEL can be described as a set of localminima connected by transition pathways, formingan undirected complex network [192]. Such networksare described by a variety of metrics, one of whichis the degree distribution, i.e. the distribution ofprobabilities of a minimum (vertex, in the languageof networks) having a certain number of connections(edges). Potential energy landscapes are found to havea power law degree distribution, a so-called “scale-free”network (a common motif in many fields that employgraph theory), and there is a strong link betweenthe distribution of basin hypervolumes in the PELand the fractal packing of Apollonian hyperspheres[193]. In contrast with a random network, which hasan exponential degree distribution, the power-law tailmeans there is a non-trivial chance of finding a vertexwith abnormally many edges, a “hub”. This featureleads to the phenomenon of small-world networks: thegeodesic between two points is shorter in scale freenetworks than for random networks, since hubs actas conduits. As an example, the degree distributionof several Lennard-Jones clusters is shown in figure 8,showing a power law degree distribution for the largerclusters. This feature is exploited in crystal structureprediction, where the hubs are generally associatedwith low energy configurations and metabasins. Thereis a strong correlation between the degree of aminimum and its potential energy (the global minimumis generally the most connected vertex) for Lennard-Jones clusters. This can be explained by the largehyperarea surrounding lower minima, which meansthere is a higher probability of finding a transitionpathway to them [192].

Beyond the degree distribution, there is a richphenomenology of complex networks that allows manydifferent analyses and many parameters that canpotentially be used to describe PELs. One exampleis the “community” structure, or clustering of hubswithin a landscape — the tendency of of local minimato aggregate in metabasins, and metabasins to formbasins of metabasins. One such study identifiedvarying degrees of community structure in Lennard-Jones clusters [194]. One feature of scale free networksis preferential attachment: when the network growsupon adding more atoms and as a result more inherent

24

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Figure 8. Cumulative degree distribution for Lennard-Jonesclusters 9-14 (Reproduced with permission from J. Chem. Phys.122, 84105 c©2005, AIP Publishing LLC [192]). k is the degreeof a graph vertex, i.e. the number of connected edges. The graphgets progressively closer to a power law distribution with slope−(γ − 1), and γ = 2.78 (straight line). Since the behaviourdoes not hold for the smaller clusters, the system is not trulyscale free, but for larger clusters, there is a power law degreedistribution.

states, the power law tail of the degree distributionarises from a tendency of new edges to be formedby preferentially attaching to verticies with a higherdegree. It transpires that this behavior can beinfluenced by varying the range of the Lennard-Jones potential by adjusting the exponents (whichare usually 12 and 6). A shorter range potential(smaller exponents) is known to result in a morecomplex, rougher PEL with more vertices and edges inits network, since the weaker interaction from distantparticles leads to a larger number of stable packings.The effect of increasing the range of the potential canbe more rigorously explained using catastrophe theory[195]. Since the number of edges increases faster thanthe number of nodes, and these new edges undergopreferential attachment, leading to a scale free network[196].

Up to this point, the construction of the networkhas dispensed with all notions of activation barriers;however, it is possible to generalise this approach for atreatment of dynamics by, for example, superimposinga master equation onto the network [197, 198]. Theproblem with this approach is defining connectivity:in order to find connections between local minima,eigenvector-following algorithms are often employed inorder to locate saddle points along the paths betweenbasins with the lowest barrier. In this context, thedefinition of a connection between two minima is notwell defined, and moreover, there is evidence to suggestthat it is incorrect to use minima of the gradient ofthe total energy, since the majority of points where

|∇E|2 = 0 are non-stationary points of the potentialenergy [172]. In light of this finding, a differentapproach may be necessary.

In a related approach, Wang et al. [127] proposethe “potential energy landscape ensemble”, which isdefined as the set of configurations for which thepotential energy is less than or equal to some constant“landscape energy”, EL. This has the effect of de-emphasising the concept of dynamics between basinsas a process of surmounting activation barriers. Inthis picture, slow dynamics below the glass transitiontemperature is a result of the long and tortuous,but effectively barrierless, routes between basins —i.e. the glass transition is dictated by entropic ratherthan energetic considerations (figure 9). This can bephysically justified by the exponential dependence ofbarrier height on the probability of a barrier transition:crossing many low barriers is far more likely thancrossing one high barrier.

In the case of an extremely low EL, it is possibleto envisage disconnected regions of the ensemble withno connected pathways, which corresponds to thetransition to the non-ergodic regime. The extensionthat configurations follow geodesic paths betweenbasins [128] may provide a more natural definitionfor network connectivity than the more commonmodel of activated dynamics. In the case of thebinary Lennard-Jones mixture, it has been shownthat as the temperature decreases, geodesics increaseas paths between minima become progressively moreconvoluted, much more so for glass-forming liquids[128]. Intriguingly, they observe the same behaviour(and successfully predict the onset of dynamicheterogeneity) for a hard sphere system, which hasno inherent states or intervening barriers [199], raisingthe question of how much of the behaviour can beattributed to dynamics unique to a hard sphere system,which differs from more realistic soft potentials in somevery important ways. The potential energy landscapeensemble paradigm has stimulated interest in NVU(constant potential energy) dynamics. Physically,NVU dynamics is a manifestation of Newton’s secondlaw: it models a system moving at constant velocitywith zero friction, such that the force is always normalto the surface and thus does no work, conservingkinetic energy. It has been shown that NVU dynamicsis equivalent to microcanonical (NVE) in the limitof large N [200], and its results are as good as forcanonical (NVT) and stochastic dynamics (such asMonte Carlo) [201].

5.6. Discussion

The general trend in the literature is to conclude thatthe glass transision can only be described in termsof non-local, collective rearrangement of atoms, rather

25

Activated process

Entropic process

Figure 9. A two-dimensional illustration of the potentialenergy landscape ensemble (adapted from reference [128]). Theensemble consists of configurations with V (R) < EL. As EL

is decreased, the forbidden regions (red) become larger, andthe paths between minima become longer and more tortuous,resulting in slow dynamics; the potential energy landscapeproblem essentially recasts an activated process as an entropicprocess.

than localised events or the static structure. There is,however, evidence to suggest that there are strong linksbetween the potential energy landscape and dynamics,and that aspects of the dynamic heterogeneity arisefrom the static structure [147].

Single-structure descriptions of glasses do not ingeneral tell the whole story. In terms of dynamicalheterogeneity, free volume models of glasses are agenerally unreliable descriptor, since free volume is asingle particle property. Similarly, describing glassesas “disordered crystals” consisting of polyhedral unitsof the crystalline structure also fail when the localcoordination in the liquid phase differs from thecrystalline phase. Voronoi tesselation offers a useful,rigorous definition of free volume that demonstratesicosahedral short-range ordering in metallic glasses,where weak orientational constraints on the firstcoordination sphere turn the glass structure in to ahard sphere packing problem rather than a chemicalbonding problem.

By introducing the PEL, we examine not onlythe static structure, but the whole ensemble of stablestructures. A transition between two points on thePEL generally involves the movement of more than oneparticle, so such an approach is capable of describingcollective motion. Just by looking at the structuraldensity of states of randomly generated structures, itcan be seen that glass-formers and crystal-formers have

very different PELs: glass-formers are characterisedby a low energy “long tail”, that is, a high densityof states close to the crystalline minimum. We canrefine this picture by additionally considering eitherentropic or activated transition pathways between thelocal minima. Thus far, most analyses of this typeon glassy systems have been primarily qualitative, sothere is a need for more work treating network modelsof glasses quantitatively. One way of achieving thisis by mapping minima and transition paths of a PELto the vertices and edges of a complex network andperforming statistical analysis on this simplified model.The converse can also be achieved, starting from anetwork model of a glass as a means of coarse-graining,and modelling a real-time trajectory between the localminima using a rate equation.

The dynamics of the system are implicit in thePEL. The dynamical matrix can be computed, which inturn can be used to determine the mechanical stabilityand properties of any configuration. Such methodsmay also be used to describe the the aging of glassysystems over timescales inaccessible to simulation usingtransition state theory, and unstable and diffusivemodes offer an alternative definition of transitionpathways. In such a way, modelling techniques thatmay be individually insufficient can be combined tooffer a better understanding of glasses through a verysimple paradigm, the PEL.

6. Model glass-forming systems

Ideally, we want to use fully transferable ab initiopotentials generated “on the fly” using densityfunctional theory (DFT) [202, 203] to characterisethe difference between good and bad glass formers.However, there is a wealth of research in the literatureemploying computationally cheap, idealised forcefieldmodels to study potential energy landscapes. In thissection, we summarise these results.

6.1. The binary Lennard-Jones (BLJ) glass

The binary Lennard-Jones mixture is one of the mostcommonly used model systems for glass formation,aptly described as ‘the “drosophila” of computationalstudies of glass-forming systems’ [122]. Classicalforcefields are computationally cheap, and allowmolecular dynamics simulations on a timescale ofmilliseconds (compared with nanoseconds for ab initiomolecular dynamics). Moreover, a judicious choiceof forcefield parameters and composition results inthe tendency towards crystallisation to be suppressed.Many studies in the literature employ a potentialsimilar to that of Kob and Andersen, a binary mixtureof Lennard-Jones particles with the same mass in an80:20 ratio [28]. The forcefield parameters were chosen

26

to be similar to amorphous Ni80P20 in Weber andStillinger’s earlier model [204] to avoid crystallisationat low temperatures. Above its melting temperature,the BLJ is a fragile liquid and is good glass former. Itis well characterised in the literature, with a knownKauzmann temperature [205], and it undergoes asecond order phase transition to a glassy state at itsmode-coupling temperature.

An explicit computation of the density of statesof this system as a function of temperature revealsthat the curve of logarithm of the density of statesbecomes steeper as the energy decreases [206]. This isnot unexpected, and confirms that as the temperaturedecreases, the number of accessible states becomessmaller. Moreover, the number of pathways betweenminima decreases as the temperature decreases, aneffect which is more pronounced for fragile systems,facilitating the super-Arrhenius behaviour of glasses.This can be linked to the cage picture, in which aparticle is trapped in a cage formed by its nearestneighbours, such that a considerable barrier mustbe overcome for it to break out [207]. The self-diffusion constant for individual atoms show Arrheniusbehaviour over short timescales, and super-Arrheniusbehaviour is recovered after averaging over all atomsover an ergodic timescale. This results not fromthe distribution of barriers, but from the negativecorrelation in atomic displacements over successivetimesteps, i.e. reversals in direction, which maybe analogous to non-productive cage-breaking (seesection 5.4) [208, 209].

In the PEL picture, relaxations on two timescalesare demonstrated by the BLJ mixture: high fre-quency α-process, corresponding to transitions be-tween minima within a metabasin, and a low-frequency β-process, corresponding to transitions be-tween metabasins. Above TMCT, the liquid is a multi-trap system, for which the barriers separating inherentstructures are small compared to the kinetic energy,and a single-trap system below it, when the barriersbecome large [210]; in this way it is possible to rigor-ously define metabasins in the context of a numericalsimulation. Metabasins can be divided into liquid- andsolid-like categories based on the “waiting time” τ ofthe configuration therein: solid-like metabasins have awaiting time τ > τ∗, where τ∗ is a crossover value,and vice versa for liquid-like metabasins [211]. This isfurther evidence that the dynamical heterogeneity mayhave a static structural origin.

For small periodic BLJ systems such as the 32atom system described in reference [83], or smallBLJ clusters, it is possible to almost completely mapthe minima of the PEL; however, such systems aresubject to finite size effects. The surface effects presentin clusters are well known, but it has been shown

that that finite size effects are insignificant for N ≥60 [212]. Since BLJ simulations are so cheap andcan be productively performed on a modern desktopcomputer, it is tempting to examine larger systems,but there is a caveat: Doliwa and Heuer demonstratethat a 130 atom BLJ system essentially behaves astwo weakly coupled 65-atom BLJ subsystems [213].If the subsystems are sufficiently weakly correlated, ittherefore makes sense to simulate only one subsystemto avoid obfuscating the information contained in thePEL [214]. Transitions between local minima arefrequently modelled as a random walk, but this turnsout not to be the case for hops between single basins;however hops between basins within a metabasinstrongly resemble a random walk [215].

A simple thermodynamic model suggests that thefragility of a liquid is proportional to the logarithm ofthe number of states [216]. Sastry uses a BLJ mixtureto establish a quantitative connection between fragilityand the PEL [217]. In addition to the multiplicityof states (encoded in the configurational entropy),the fragility also depends on the variance of thedistribution of states and the variation of vibrationalfrequencies (i.e. entropy) between the lowest andhighest basins.

A phenomenological observation associated withthe fragile BLJ liquid is the tendency for correlatedrearrangements of clusters at temperatures aboveTMCT which are accessible by molecular dynamics.Particle displacements appear to happen along string-like clusters [118, 218]. In a glass, most atomic motiontakes the form of small amplitude vibrations, but thestructure is jammed on a diffusive timescale. A regionof jammed molecules can only become mobile if it isadjacent to an unjammed region, a process which formsdistinct chains [219].

Recently, a new molecular dynamics techniqueusing a BLJ model was used to attempt to generatea model glass with the characteristics of ultrastableorganic and bulk metallic glasses synthesised by vapourdeposition. Groups of particles were introduced toa simulation cell some distance above a constrainedsubstrate layer, followed by an equilibration phaseand finally an energy minimisation. The resultingglass was found to have the properties of a vapourdeposited equilibrium supercooled liquid, and it wasnoted that the optimal temperature for preparation ofthese ultrastable glasses corresponds to the Kauzmanntemperature [10]. It is notable that the radialdistribution function of this model ultrastable glassis indistinguishable from an ordinary quench glass;instead, the stability has been explained in terms of thecage picture. In a bulk glass, the motion of a particle isstrongly constrained by the cage formed by its nearestneighbours; however, on the surface, it is free to “roll”

27

Figure 10. Schematic of the effects of cooling rate versusvapour deposition on glass stability (Reprinted by permissionfrom Macmillan Publishers Ltd: Nature Mater. 12, 94, c©2013[220]). When a liquid is cooled, it falls out of equilibrium at atemperature that depends on the cooling rate. Lower coolingrates allow the system to explore lower energy metabasins, whilevapour deposition and the molecular dynamics technique ofParisi et al. make extremely deep and narrow minima accessible.

with a reduced free energy cost, enhancing its mobilityand allowing a more extensive exploration of the PEL[220] (figure 10). It should, however, be noted thatthis pciture is at odds with experiments suggestingthat ultrastable metallic glasses are not necessarily thelowest thermodynamic state, but are kinetically verystable [9]. The question of whether ultrastable glassesare in fact the most thermodyanmically stable remainsan open question in the literature.

As a final note, ring-polymer molecular dynamics(RPMD), which goes beyond Born-Oppenheimermolecular dynamics and simulates nuclei as quantummechanical particles, has been used in conjunction witha BLJ mixture [221]. The degree of “quantumness”is controlled by the Λ∗ parameter, which is theratio of the de Broglie wavelength to the particlesize. Interestingly, the quantum mechanical glass ismuch less structured than the classical, since quantummechanical tunneling across PEL barriers is permitted.Moreover, the diffusivity decreases as a function of Λ∗

until the size of the quantum wave packet becomescomparable to the radius of the cage enclosing theparticle in the glass (at approximately Λ∗ = 1, i.e. a deBroglie wavelength comparable to the particle size), atwhich point it can tunnel through the cage, resultingin a sharp increase in diffusivity. Whilst this is a niceillustration of a theoretical quantum glass, this effect isunlikely to affect a real glass, for which the temperatureregime will be too high, and the atoms too massiveunless it contains hydrogen. More recent calculationssuggest that quantum effects tend to broaden the glasstransition regime [222].

6.2. Amorphous silica

Silica (SiO2) is important not only because ofits ubiquity in natural and man-made amorphousmaterials, but also because it is an extreme case ofa strong liquid as can be seen by its position on theAngell plot (figure 1). Molecular dynamics simulationsat varying pressures demonstrate that silica is a strongliquid at ambient pressure, when the silicon atoms arelargely four-coordinated, but as the pressure increases,they become five- and six-coordinated, signalling theonset of fragility [223]; this was recently confirmedexperimentally [224].

The availability of a reliable classical potentialthat has been validated against ab initio models, thevan Beest-Kramer-van Santen (BKS) potential [225],means it has been very well studied. A study on thePEL topology of BKS silica suggests a characteristictemperature Tc =3500 K, above which the systemexplores anharmonic basins, but below which there isno sign of anharmonicity [226]. Silica is also notable forits transition from a fragile liquid to a strong liquid asthe temperature decreases below a crossover point, Tc,which is found to be around 3300 K [227] for the BKSmodel. This transition has been attributed to a lowenergy cutoff in the density of structural states, whichcoincides with the vanishing of the number of three-and five-coordination defects [228]. The similarity ofthe two aforementioned crossover temperatures suggestthat there is a possible connection between basinanharmonicity and liquid fragility. Tc coincides witha peak in the specific heat CV , the maximum densityand a maximum in the isothermal compressibility KT ,phenomena which are associated with a polyamorphictransition in glasses, i.e. the transition from a low-density glass to a high-density glass. This suggeststhat the fragile-strong transition is associated with apolyamorphic transition, and moreover that all strongliquids are candidates for polyamorphism [229]. Thiscan be physically justified by the existence of twodifferent arrangements of particles in tetrahedrallycoordinated materials resulting in two characteristicinterparticle distances. For example, low densityice Ih and the compact high pressure phases in thecase of water [37], which also displays a fragile-strong liquid transition [230]. Kushima et al. [231]extend this to all viscous liquids, suggesting thatthere are two crossover temperatures for all glassformers: upon cooling from high temperature, thereis a strong to fragile transition, followed by a fragileregime, and finally a fragile to strong transition. Thedifference between fragile and strong liquids lies in therearrangement processes that occur between basins.Bond switching processes have high activation barriersprevalent at high temperatures, compared withprocesses involving dangling bonds which dominate at

28

low temperatures. In the cage entrapment paradigm,high-barrier mechanisms correspond to cage-breaking,while low barrier mechanisms are related to theredistribution of free volume [207]. However, thebarrier height is in general independent of temperature,except in a limited “intermediate” temperature range,over which the barrier changes with temperature, andthe liquid is fragile [231].

6.3. Hard sphere model

A system of hard spheres is arguably the simplestsystem that displays fluid-solid coexistence and a glasstransition. Hard spheres are essentially particles withshort range repulsive interactions, and without longerrange dispersive interactions; they do not interactunless they overlap. A hard sphere model places theemphasis on efficient packing rather than chemicalbonding, and is often used to describe the glasstransition as a “jamming” transition as in the caseof granular materials, the point at which a systemdevelops a yield stress in a disordered state [232].

This model has a single control parameter, thedimensionless density n∗ = ρ0σ

3, where ρ0 is theaverage density in the crystalline phase and σ is thehard sphere diameter. Increasing n∗ has the sameeffect as increasing the temperature [97]. A hard spheresystem will remain structurally disordered if cooledfast enough, and under further compression it will jaminto one of a multitude of possible packings with aminimal free volume, reaching the limit of “randomclose packing” at an infinite compression rate [233].

A study on devitrification (crystallisation from aglassy or amorphous phase) shows in order to formcrystallites, particles undergo a short range “shuffling”motion, for which the mean displacement is lessthan a diameter — i.e. crystallisation occurs in spiteof the arrest of diffusive motion [234]. The hardsphere free energy landscape undergoes a pressure-induced ‘roughness transition’, at which point basinsbecome aggregates of sub-basins in a fractal manner(figure 11), representing a multitude of marginallystable states [235]. Expanding on these interestingresults using more realistic models may eventuallyreveal the microscopic root of glass-forming ability incomparison with crystal-forming ability.

O’Hern et al. demonstrate that a hard spheresystem undergoes a jamming transition, defined asthe point at which the bulk and shear modulisimultaneously become non-zero, at a critical density,and proceed to show that this transition point has avariety of interesting properties, such as a divergingradial distribution function and an excess of low-frequency normal modes. [232]. This supports thenotion of cooperatively rearranging regions, since softdegrees of freedom are collective. Moreover, fewer and

Low LowHigh High

e

Figure 11. The pressure-induced transition to a fractal freeenergy landscape for hard spheres (Reprinted by permission fromMacmillan Publishers Ltd: Nature Comm. 5, 3725 c©2014 [235]).As the control parameter (packing fraction ϕ) is increased aboveits glass transition point ϕg , basins divide fractally into sub-basins and sub-sub-basins representing marginally mechanicallystable states close to a jamming transition.

fewer modes participate in structural rearrangementas the density approaches its critical value, perhapssuggesting cooperative regions that are growing in size[236].

At first glance, the hard sphere model appears tobe somewhat academic, since real systems always havelong range attractive forces. It does, however, remain auseful model, since all realistic systems have a distinctshort-range repulsion cutoff, such as the minimum of a6-12 Lennard-Jones potential. Moreover, at the densityof solids, the primary effect of long-range attractionsis to maintain the density of the solid such that therepulsions can have an effect [232]. Moreover, hardspheres are surprisingly good at explaining the glass-forming ability of bulk metallic glasses, which arecharacterised by weak, non-directional interactions. Ina binary hard sphere system, the best glass formershave an atomic size ratio that compromises betweentwo competing pressures: minimising it to improve thepacking efficiency, and maximising it to prevent phaseseparation [233].

A comprehensive review of hard sphere glasses canbe found in reference [237].

6.4. Discussion

We have described model systems that describe verydifferent types of glasses. On one hand, we have abinary Lennard-Jones model of amorphous silica, anexample of a strong liquid that forms a glass withbonding network with high covalent character. Atthe other end of the spectrum, hard spheres are anidealisation of metallic glasses, derived from fragileliquids which lack the orientational constraints ofstrong liquids. The structure of such glasses becomes aquestion of filling space with spheres of different sizes,

29

and a mechanical jamming transition, as opposed tothe structure of a more rigid covalent bonding network.

The binary Lennard-Jones mixture is a goodgeneral choice of model system, since its glass-formingability can be controlled via the composition, andits fragility can be controlled either by adjustingthe hardness of the potential, or changing theparticle density, which has essentially the same effect.Moreover, it is well characterised in the literature, andits mode-coupling temperature has been calculated forcertain compositions and parameterisations.

An important point that arises from these cheapcalculations is the effect of system size. One mightnaıvely assume that in order to capture medium-rangeordering, it is necessary, due to the lack of symmetry,to use the largest supercell possible. This is notnecessarily the case, as it has been demonstrating thatonce a critical system size is exceeded, the supercell canbe decomposed in to weakly interacting subsystems,the effect of which may mask the properties of realinterest.

7. Ab initio models

In addition to providing a more reliable and trans-ferable structural model, ab initio methods are capa-ble of describing electronic structure. Just as trans-lational symmetry in crystals gives rise to Bloch elec-tronic states which are delocalised, disorder in amor-phous solids gives rise to localised states that canbe more naturally described using Wannier functions[238]. Tight binding calculations on amorphous siliconmanifest electronic states which appear as sparse buthighly localised “islands” of electron density, and arecaused by rare, major distortions in the local structure.These are often connected by “filaments” [239]. Thesestructural filaments are related to the Urbach tail inamorphous silicon, the exponentially decaying densityof states at the band edge [240, 241]. There is also astrong correlation between the degree of electronic lo-calisation and the sensitivity of the electrons to latticevibrations, i.e. strong localisation is associated with alarger electron-phonon coupling, as has been demon-strated for hydrogenated amorphous silicon [242, 243].

In fact, the paucity of ab initio calculationson glasses reflects a general neglect of the effect ofelectronic structure on glass forming ability. There isevidence that glass forming ability is correlated withthe electronic density of states. Al is a commoncomponent of bulk metallic glasses as a dopant,although it is rare to find Al-rich (>30%) glasses. Ithas been suggested that this is because Al has morefree electrons (3) than most other metals, enhancingthe density of states at the Fermi level and reducingthe free energy of the amorphous state [244].

Icosahedral short range ordering is favoured bynon-directional metallic bonds, but electronic structurecalculations show that Pd40Ni40P20 achieves a superiorpacking ratio via a “hybrid” scheme involvingboth metal-centred icosahedra and highly directionalcovalently bound metalloid-centred tricapped trigonalprisms [245]. This is further evidence of the importanceof electronic structure in glass-formation, since densityfunctional theory is capable of transferably modellingglasses formed from both strong and fragile liquids.

A note of caution is in order when it comes to usinglocal and semilocal DFT in the context of amorphousstructures. When a defect is introduced into a crystalstructure, it can introduce states into the band gap.These are manifested physically as localised states.Since glasses are very structurally disordered, thereare many of these localised states, and in generallocal and semilocal exchange-correlation functionalswill fail to capture the non-locality and correlations.This effect can be mitigated using hybrid exchange-correlation functionals, but in the absence of any formof symmetry, these will be extremely expensive. Onewould expect to see qualitative as well as quantitativechanges in the electronic structure as a result of thiseffect, which can have significant knock-on effects.

8. Glass-forming ability versus crystal-formingability

Glass forming ability is a difficult quantity tocharacterise experimentally, since crystal nucleationdepends on a variety of factors including temperature,pressure and composition, not to mention externalfactors such as chemical and physical impurities,container wall effects and shear flow condition inthe liquid. To date, glass-forming ability has beendescribed somewhat phenomenologically, for example,in the form of time-temperature-transformation curves(TTT) [246, 247]; other empirical formulae usethe liquidus, crystallisation and glass transitiontemperatures to derive a parameter capable ofpredicting whether a bulk metallic glass has a highglass forming ability [13–15].

Experimentally, it has been possible to mapthe glass-forming ability of a 5-component alloy incomposition space. Glass forming ability is controlledby two competing thermally activated processes:the Gibbs free energy barrier to the formationof a critical nucleus W and the barrier to theconfigurational rearrangement of the liquid ∆G. Thiscan be summarised in the following expression forthe characteristic nucleation time τα of the crystallinephase α [248].

τα = η−1 exp1

kT[W (T, c) + ∆Gα(T, c)] (55)

30

Willow Palm Banyan

Figure 12. Disconnectivity graphs corresponding to threearchetypal PELs (adapted from reference [251])

Here, the Gibbs free energy barriers W and ∆G arefunctions of temperature and composition (c), and η isa transition attempt frequency.

Whilst there are many empirical guidelines fordetermining the glass-forming ability of compoundssuch as bulk metallic glasses, a microscopic rationaleremains lacking [249]. Recent molecular dynamicssimulations suggest a significant difference in theliquid-crystal interface between good and poor glassformers suggesting that a difference in heterogeneoussusceptibilities may be responsible for the difference[250]. They can also be distinguished qualitativelyby the “shape” of their respective PELs, althoughhow to visualise such a high-dimensional space isa challenge in itself. Despa et al. achieve this bycategorising disconnectivity graphs into three distinctmotifs (figure 12): “weeping willow”, in which thebarriers are large compared with the relative energiesof the basins, “palm tree”, in which the barriersare small compared with the relative energies of thebasins, and “banyan tree”, which is characterisedby many degenerate minima[251]. A glass formerrequires sufficiently high barriers below Tg, and a highdegeneracy near the global minimum. In principle, thebanyan and possibly willow tree motifs might be usedto identify good glass formers.

The identification of compositions that wouldmake good bulk metallic glasses used to be a timeconsuming trial and error procedure in which onecomposition could be generated per day. Recentdevelopments have made the searching of compositionspace much more efficient [252], but gaps in thephysical justification remain, and a theoretical methodfor generating glass forming compositions wouldstreamline the process considerably.

9. Conclusions and perspectives

Over the course of this review, we have made thecase that the PEL is a valid and useful paradigmfor understanding glass formation. The dynamicsof the system slow beyond any experimental, letalone computational timescale when the temperatureis lowered to the point when glass transition

intervenes and the system falls out of equilibrium.Thus stochastic sampling methods such as stochasticquenching and Monte Carlo are necessary to probe theenergy landscape below the transition temperature.

Much of the work on energy landscapes has em-ployed computationally cheap, and specialised or non-transferable models (in particular, binary Lennard-Jones and hard sphere systems), limiting the predic-tive power of the PEL. Sampling using relatively ex-pensive ab initio techniques would facilitate theoreticalpredictions; calculations on this scale are certainly pos-sible now, for periodic systems large enough to avoidfinite size and small enough to avoid decoupling intonon-interacting subsystems (around 60 atoms). This istrue for atomic glasses, including bulk metallic glassesand amorphous thin films synthesised via vapour de-position, but perhaps not for more complex molecu-lar, polymeric and organic glasses yet. These ab initiomodels can in turn be used to generate coarse-grainedmodels using activated dynamics and graph theory tofurther improve predictive power.

A criticism of PEL analysis is that it paintsa largely static picture of glasses, and is incapableof describing the glass transition via, for example,a dynamic heterogeneity or a space-time phasetransition. It can give us a wealth of information on thethermodynamically stable and metastable structure,and transitions between them, but can it tell usanything about the transition from a supercooled liquidto a glass? PEL analysis allows the developmentof structural models of glasses, but a static analysisis not likely to be sufficient to describe the glasstransition — for example the static correlations areessentially identical between liquids and glasses, and noconvincing connection has been made between diffusionand free volume in glass-forming liquids. However,there is evidence that the dynamic hetereogeneity mayhave a structural origin. One example is the correlationof the low frequency modes of a system with thepropensity of particles to undergo large displacements;another is the observation of critical-like fluctuationsin static structural order near the glass transition.Normal modes in particular are routinely derived fromstatic structures via phonon calculations, and theirapplication to more realistic models could be a fruitfularea of research.

There are two key areas in which PEL analysisis likely to prove particularly useful: firstly in thegeneration of high quality structural models of glasses,and secondly in the identification of good glass-forming compositions of metallic and vapour-depositedthin film glasses. The former can be achievedthrough some variation of stochastic quenching, andthe latter through analysis of the density of statesand network topology of the DFT energy landscape

REFERENCES 31

of real materials. At a fundamental level, it offers anattractive route to understanding the microscopic basisof glass formation.

Acknowledgments

This work was financially supported by the SwedishFoundation for Strategic Research (SSF) synergyprogram FUNCASE. Z. R. thanks Nuala M. Caffrey,Chris J. Pickard, Olle Hellman and Nina Shulumbafor helpful discussions. B. A. acknowledges financialsupport from the Swedish Research Council (VR) grantNo. 621-2011-4417. I. A. A. acknowledges support fromSSF program SRL (Grant No. 10-0026, VR grant No.621-2011-4426), the Russian Federation Ministry forScience and Education (grant No. 14.Y26.31.0005) andthe Tomsk State University Academic D. I. Mendeleevfunding program.

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